907 57 5MB
English Pages [580] Year 2017
A Graduate Course in Applied Cryptography Dan Boneh and Victor Shoup
Version 0.3, December 2016
Preface Cryptography is an indispensable tool used to protect information in computing systems. It is used everywhere and by billions of people worldwide on a daily basis. It is used to protect data at rest and data in motion. Cryptographic systems are an integral part of standard protocols, most notably the Transport Layer Security (TLS) protocol, making it relatively easy to incorporate strong encryption into a wide range of applications. While extremely useful, cryptography is also highly brittle. The most secure cryptographic system can be rendered completely insecure by a single specification or programming error. No amount of unit testing will uncover a security vulnerability in a cryptosystem. Instead, to argue that a cryptosystem is secure, we rely on mathematical modeling and proofs to show that a particular system satisfies the security properties attributed to it. We often need to introduce certain plausible assumptions to push our security arguments through. This book is about exactly that: constructing practical cryptosystems for which we can argue security under plausible assumptions. The book covers many constructions for di↵erent tasks in cryptography. For each task we define a precise security goal that we aim to achieve and then present constructions that achieve the required goal. To analyze the constructions, we develop a unified framework for doing cryptographic proofs. A reader who masters this framework will be capable of applying it to new constructions that may not be covered in the book. Throughout the book we present many case studies to survey how deployed systems operate. We describe common mistakes to avoid as well as attacks on realworld systems that illustrate the importance of rigor in cryptography. We end every chapter with a fun application that applies the ideas in the chapter in some unexpected way.
Intended audience and how to use this book The book is intended to be self contained. Some supplementary material covering basic facts from probability theory and algebra is provided in the appendices. The book is divided into three parts. The first part develops symmetric encryption which explains how two parties, Alice and Bob, can securely exchange information when they have a shared key unknown to the attacker. The second part develops the concepts of publickey encryption and digital signatures, which allow Alice and Bob to do the same, but without having a shared, secret key. The third part is about cryptographic protocols, such as protocols for user identification, key exchange, and secure computation. A beginning reader can read though the book to learn how cryptographic systems work and why they are secure. Every security theorem in the book is followed by a proof idea that explains at a high level why the scheme is secure. On a first read one can skip over the detailed proofs
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without losing continuity. A beginning reader may also skip over the mathematical details sections that explore nuances of certain definitions. An advanced reader may enjoy reading the detailed proofs to learn how to do proofs in cryptography. At the end of every chapter you will find many exercises that explore additional aspects of the material covered in the chapter. Some exercises rehearse what was learned, but many exercises expand on the material and discuss topics not covered in the chapter.
Status of the book The current draft only contains part I and the first half of part II. The remaining chapters in parts II and part III are forthcoming. We hope you enjoy this writeup. Please send us comments and let us know if you find typos or mistakes. Citations: While the current draft is mostly complete, we still do not include citations and references to the many works on which this book is based. Those will be coming soon and will be presented in the Notes section at the end of every chapter.
Dan Boneh and Victor Shoup December, 2016
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Contents 1 Introduction 1.1 Historic ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Terminology used throughout the book . . . . . . . . . . . . . . . . . . . . . . . . . .
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Secret key cryptography
2 Encryption 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shannon ciphers and perfect security . . . . . . . . . . . . . . . 2.2.1 Definition of a Shannon cipher . . . . . . . . . . . . . . 2.2.2 Perfect security . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The bad news . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Computational ciphers and semantic security . . . . . . . . . . 2.3.1 Definition of a computational cipher . . . . . . . . . . . 2.3.2 Definition of semantic security . . . . . . . . . . . . . . 2.3.3 Connections to weaker notions of security . . . . . . . . 2.3.4 Consequences of semantic security . . . . . . . . . . . . 2.3.5 Bit guessing: an alternative characterization of semantic 2.4 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Negligible, superpoly, and polybounded functions . . . 2.4.2 Computational ciphers: the formalities . . . . . . . . . . 2.4.3 Efficient adversaries and attack games . . . . . . . . . . 2.4.4 Semantic security: the formalities . . . . . . . . . . . . . 2.5 A fun application: anonymous routing . . . . . . . . . . . . . . 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stream ciphers 3.1 Pseudorandom generators . . . . . . . . . . . . . . . . 3.1.1 Definition of a pseudorandom generator . . . . 3.1.2 Mathematical details . . . . . . . . . . . . . . . 3.2 Stream ciphers: encryption with a PRG . . . . . . . . 3.3 Stream cipher limitations: attacks on the one time pad 3.3.1 The twotime pad is insecure . . . . . . . . . .
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3.3.2 The onetime pad is malleable . . . . . . . . . . . . . . . Composing PRGs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 A parallel construction . . . . . . . . . . . . . . . . . . . . 3.4.2 A sequential construction: the BlumMicali method . . . 3.4.3 Mathematical details . . . . . . . . . . . . . . . . . . . . . The next bit test . . . . . . . . . . . . . . . . . . . . . . . . . . . Case study: the Salsa and ChaCha PRGs . . . . . . . . . . . . . Case study: linear generators . . . . . . . . . . . . . . . . . . . . 3.7.1 An example cryptanalysis: linear congruential generators 3.7.2 The subset sum generator . . . . . . . . . . . . . . . . . . Case study: cryptanalysis of the DVD encryption system . . . . Case study: cryptanalysis of the RC4 stream cipher . . . . . . . 3.9.1 Security of RC4 . . . . . . . . . . . . . . . . . . . . . . . . Generating random bits in practice . . . . . . . . . . . . . . . . . A broader perspective: computational indistinguishability . . . . 3.11.1 Mathematical details . . . . . . . . . . . . . . . . . . . . . A fun application: coin flipping and commitments . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Block ciphers 4.1 Block ciphers: basic definitions and properties . . . . . . . . . . . . . . . . . 4.1.1 Some implications of security . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Efficient implementation of random permutations . . . . . . . . . . . 4.1.3 Strongly secure block ciphers . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Using a block cipher directly for encryption . . . . . . . . . . . . . . 4.1.5 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Constructing block ciphers in practice . . . . . . . . . . . . . . . . . . . . . 4.2.1 Case study: DES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Exhaustive search on DES: the DES challenges . . . . . . . . . . . . 4.2.3 Strengthening ciphers against exhaustive search: the 3E construction 4.2.4 Case study: AES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Sophisticated attacks on block ciphers . . . . . . . . . . . . . . . . . . . . . 4.3.1 Algorithmic attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Sidechannel attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Faultinjection attacks on AES . . . . . . . . . . . . . . . . . . . . . 4.3.4 Quantum exhaustive search attacks . . . . . . . . . . . . . . . . . . . 4.4 Pseudorandom functions: basic definitions and properties . . . . . . . . . . 4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Efficient implementation of random functions . . . . . . . . . . . . . 4.4.3 When is a secure block cipher a secure PRF? . . . . . . . . . . . . . 4.4.4 Constructing PRGs from PRFs . . . . . . . . . . . . . . . . . . . . . 4.4.5 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Constructing block ciphers from PRFs . . . . . . . . . . . . . . . . . . . . . 4.6 The tree construction: from PRGs to PRFs . . . . . . . . . . . . . . . . . . 4.6.1 Variable length tree construction . . . . . . . . . . . . . . . . . . . . v
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The ideal cipher model . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Formal definitions . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Exhaustive search in the ideal cipher model . . . . . . . . 4.7.3 The EvenMansour block cipher and the EX construction 4.7.4 Proof of the EvenMansour and EX theorems . . . . . . . 4.8 Fun application: comparing information without revealing it . . . 4.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Chosen Plaintext Attack 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Security against multikey attacks . . . . . . . . . . . . . . . . . . 5.3 Semantic security against chosen plaintext attack . . . . . . . . . . 5.4 Building CPA secure ciphers . . . . . . . . . . . . . . . . . . . . . . 5.4.1 A generic hybrid construction . . . . . . . . . . . . . . . . . 5.4.2 Randomized counter mode . . . . . . . . . . . . . . . . . . 5.4.3 CBC mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Case study: CBC padding in TLS 1.0 . . . . . . . . . . . . 5.4.5 Concrete parameters and a comparison of counter and CBC 5.5 Noncebased encryption . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Noncebased generic hybrid encryption . . . . . . . . . . . . 5.5.2 Noncebased Counter mode . . . . . . . . . . . . . . . . . . 5.5.3 Noncebased CBC mode . . . . . . . . . . . . . . . . . . . . 5.6 A fun application: revocable broadcast encryption . . . . . . . . . 5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Message integrity 6.1 Definition of a message authentication code . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 MAC verification queries do not help the attacker . . . . . . . . . . . . . . . . . . . 6.3 Constructing MACs from PRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Prefixfree PRFs for long messages . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The CBC prefixfree secure PRF . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 The cascade prefixfree secure PRF . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Extension attacks: CBC and cascade are insecure MACs . . . . . . . . . . . 6.5 From prefixfree secure PRF to fully secure PRF (method 1): encrypted PRF . . . 6.5.1 ECBC and NMAC: MACs for variable length inputs . . . . . . . . . . . . . 6.6 From prefixfree secure PRF to fully secure PRF (method 2): prefixfree encodings 6.6.1 Prefix free encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 From prefixfree secure PRF to fully secure PRF (method 3): CMAC . . . . . . . . 6.8 Converting a blockwise PRF to bitwise PRF . . . . . . . . . . . . . . . . . . . . . 6.9 Case study: ANSI CBCMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Case study: CMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 PMAC: a parallel MAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 A fun application: searching on encrypted data . . . . . . . . . . . . . . . . . . . . vi
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6.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7 Message integrity from universal hashing 7.1 Universal hash functions (UHFs) . . . . . . . . . . . . . . . . . . . 7.1.1 Multiquery UHFs . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Mathematical details . . . . . . . . . . . . . . . . . . . . . . 7.2 Constructing UHFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Construction 1: UHFs using polynomials . . . . . . . . . . 7.2.2 Construction 2: CBC and cascade are computational UHFs 7.2.3 Construction 3: a parallel UHF from a small PRF . . . . . 7.3 PRF(UHF) composition: constructing MACs using UHFs . . . . . 7.3.1 Using PRF(UHF) composition: ECBC and NMAC security 7.3.2 Using PRF(UHF) composition with polynomial UHFs . . . 7.3.3 Using PRF(UHF) composition: PMAC0 security . . . . . . 7.4 The CarterWegman MAC . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Using CarterWegman with polynomial UHFs . . . . . . . . 7.5 Noncebased MACs . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Secure noncebased MACs . . . . . . . . . . . . . . . . . . . 7.6 Unconditionally secure onetime MACs . . . . . . . . . . . . . . . . 7.6.1 Pairwise unpredictable functions . . . . . . . . . . . . . . . 7.6.2 Building unpredictable functions . . . . . . . . . . . . . . . 7.6.3 From PUFs to unconditionally secure onetime MACs . . . 7.7 A fun application: timing attacks . . . . . . . . . . . . . . . . . . . 7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Message integrity from collision resistant hashing 8.1 Definition of collision resistant hashing . . . . . . . . . . . . . . 8.1.1 Mathematical details . . . . . . . . . . . . . . . . . . . . 8.2 Building a MAC for large messages . . . . . . . . . . . . . . . . 8.3 Birthday attacks on collision resistant hash functions . . . . . . 8.4 The MerkleDamg˚ ard paradigm . . . . . . . . . . . . . . . . . . 8.4.1 Joux’s attack . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Building Compression Functions . . . . . . . . . . . . . . . . . 8.5.1 A simple but inefficient compression function . . . . . . 8.5.2 DaviesMeyer compression functions . . . . . . . . . . . 8.5.3 Collision resistance of DaviesMeyer . . . . . . . . . . . 8.6 Case study: SHA256 . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Other MerkleDamg˚ ard hash functions . . . . . . . . . . 8.7 Case study: HMAC . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Security of twokey nest . . . . . . . . . . . . . . . . . . 8.7.2 The HMAC standard . . . . . . . . . . . . . . . . . . . 8.7.3 DaviesMeyer is a secure PRF in the ideal cipher model 8.8 The Sponge Construction and SHA3 . . . . . . . . . . . . . . . 8.8.1 The sponge construction . . . . . . . . . . . . . . . . . . vii
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8.8.2 Case study: SHA3, SHAKE256, and SHAKE512 . . . . . . . . Merkle trees: using collision resistance to prove database membership Key derivation and the random oracle model . . . . . . . . . . . . . . 8.10.1 The key derivation problem . . . . . . . . . . . . . . . . . . . . 8.10.2 Random oracles: a useful heuristic . . . . . . . . . . . . . . . . 8.10.3 Random oracles: safe modes of operation . . . . . . . . . . . . 8.10.4 The leftover hash lemma . . . . . . . . . . . . . . . . . . . . . . 8.10.5 Case study: HKDF . . . . . . . . . . . . . . . . . . . . . . . . . Security without collision resistance . . . . . . . . . . . . . . . . . . . 8.11.1 Second preimage resistance . . . . . . . . . . . . . . . . . . . . 8.11.2 Randomized hash functions: target collision resistance . . . . . 8.11.3 TCR from 2ndpreimage resistance . . . . . . . . . . . . . . . . 8.11.4 Using target collision resistance . . . . . . . . . . . . . . . . . . A fun application: an efficient commitment scheme . . . . . . . . . . . Another fun application: proofs of work . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Authenticated Encryption 9.1 Authenticated encryption: definitions . . . . . . . . . . . . . . . . . . . . . 9.2 Implications of authenticated encryption . . . . . . . . . . . . . . . . . . . . 9.2.1 Chosen ciphertext attacks: a motivating example . . . . . . . . . . . 9.2.2 Chosen ciphertext attacks: definition . . . . . . . . . . . . . . . . . . 9.2.3 Authenticated encryption implies chosen ciphertext security . . . . . 9.3 Encryption as an abstract interface . . . . . . . . . . . . . . . . . . . . . . . 9.4 Authenticated encryption ciphers from generic composition . . . . . . . . . 9.4.1 EncryptthenMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 MACthenencrypt is not generally secure: padding oracle attacks on 9.4.3 More padding oracle attacks. . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Secure instances of MACthenencrypt . . . . . . . . . . . . . . . . . 9.4.5 EncryptthenMAC or MACthenencrypt? . . . . . . . . . . . . . . 9.5 Noncebased authenticated encryption with associated data . . . . . . . . . 9.6 One more variation: CCAsecure ciphers with associated data . . . . . . . . 9.7 Case study: Galois counter mode (GCM) . . . . . . . . . . . . . . . . . . . 9.8 Case study: the TLS 1.3 record protocol . . . . . . . . . . . . . . . . . . . . 9.9 Case study: an attack on nonatomic decryption in SSH . . . . . . . . . . . 9.10 Case study: 802.11b WEP, a badly broken system . . . . . . . . . . . . . . 9.11 Case study: IPsec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.12 A fun application: private information retrieval . . . . . . . . . . . . . . . . 9.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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310 311 311 311 314 319 320 322 323 323 324 325 327 330 330 330 331
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338 339 341 341 342 344 346 347 348 350 353 354 358 358 360 361 364 366 368 371 376 376 376
II
Public key cryptography
383
10 Public key tools 10.1 A toy problem: anonymous key exchange . . . . . . . . . . . . . . . . 10.2 Oneway trapdoor functions . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Key exchange using a oneway trapdoor function scheme . . . . 10.2.2 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . 10.3 A trapdoor permutation scheme based on RSA . . . . . . . . . . . . . 10.3.1 Key exchange based on the RSA assumption . . . . . . . . . . 10.3.2 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . 10.4 DiffieHellman key exchange . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 The key exchange protocol . . . . . . . . . . . . . . . . . . . . 10.4.2 Security of DiffieHellman key exchange . . . . . . . . . . . . . 10.5 Discrete logarithm and related assumptions . . . . . . . . . . . . . . . 10.5.1 Random selfreducibility . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Mathematical details . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Collision resistant hash functions from numbertheoretic primitives . . 10.6.1 Collision resistance based on DL . . . . . . . . . . . . . . . . . 10.6.2 Collision resistance based on RSA . . . . . . . . . . . . . . . . 10.7 Attacks on the anonymous DiffieHellman protocol . . . . . . . . . . . 10.8 Merkle puzzles: a partial solution to key exchange using block ciphers 10.9 Fun application: Pedersen commitments . . . . . . . . . . . . . . . . . 10.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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385 . 385 . 386 . 387 . 388 . 389 . 391 . 391 . 392 . 393 . 393 . 394 . 397 . 398 . 400 . 400 . 401 . 403 . 404 . 405 . 405 . 406
11 Public key encryption 11.1 Two further example applications . . . . . . . . . . . . . . . . . . 11.1.1 Sharing encrypted files . . . . . . . . . . . . . . . . . . . . 11.1.2 Key escrow . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Mathematical details . . . . . . . . . . . . . . . . . . . . . 11.3 Implications of semantic security . . . . . . . . . . . . . . . . . . 11.3.1 The need for randomized encryption . . . . . . . . . . . . 11.3.2 Semantic security against chosen plaintext attack . . . . . 11.4 Encryption based on a trapdoor function scheme . . . . . . . . . 11.4.1 Instantiating ETDF with RSA . . . . . . . . . . . . . . . . 11.5 ElGamal encryption . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Semantic security of ElGamal in the random oracle model 11.5.2 Semantic security of ElGamal without random oracles . . 11.6 Threshold decryption . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Shamir’s secret sharing scheme . . . . . . . . . . . . . . . 11.6.2 ElGamal threshold decryption . . . . . . . . . . . . . . . 11.7 Fun application: oblivious transfer from DDH . . . . . . . . . . . 11.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ix
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414 415 415 415 416 417 418 418 419 421 424 425 426 428 431 433 435 438 438 438
12 Chosen ciphertext secure public key encryption 12.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Understanding CCA security . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 CCA security and ciphertext malleability . . . . . . . . . . . . 12.2.2 CCA security vs authentication . . . . . . . . . . . . . . . . . . 12.2.3 CCA security and key escrow . . . . . . . . . . . . . . . . . . . 12.2.4 Encryption as an abstract interface . . . . . . . . . . . . . . . . 12.3 CCAsecure encryption from trapdoor function schemes . . . . . . . . 0 12.3.1 Instantiating ETDF with RSA . . . . . . . . . . . . . . . . . . . 12.4 CCAsecure ElGamal encryption . . . . . . . . . . . . . . . . . . . . . 12.4.1 CCA security for basic ElGamal encryption . . . . . . . . . . . 12.5 CCA security from DDH without random oracles . . . . . . . . . . . . 12.6 CCA security via a generic transformation . . . . . . . . . . . . . . . . 12.6.1 A generic instantiation . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 A concrete instantiation with ElGamal . . . . . . . . . . . . . . 12.7 CCAsecure publickey encryption with associated data . . . . . . . . 12.8 Case study: PKCS1, OAEP, OAEP+, and SAEP . . . . . . . . . . . . 12.8.1 Padding schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.2 PKCS1 padding . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.3 Bleichenbacher’s attack on the RSAPKCS1 encryption scheme 12.8.4 Optimal Asymmetric Encryption Padding (OAEP) . . . . . . . 12.8.5 OAEP+ and SAEP+ . . . . . . . . . . . . . . . . . . . . . . . 12.9 Fun application: sealed bid auctions . . . . . . . . . . . . . . . . . . . 12.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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445 . 445 . 447 . 447 . 448 . 449 . 450 . 452 . 457 . 458 . 458 . 463 . 470 . 475 . 475 . 477 . 478 . 479 . 479 . 480 . 483 . 485 . 486 . 486 . 486
13 Digital signatures 13.1 Definition of a digital signature . . . . . . . . . . . . . . . . . . 13.1.1 Secure signatures . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Mathematical details . . . . . . . . . . . . . . . . . . . . 13.2 Extending the message space with collision resistant hashing . 13.2.1 Extending the message space using TCR functions . . . 13.3 Signatures from trapdoor permutations: the full domain hash . 13.3.1 Signatures based on the RSA trapdoor permutation . . 13.4 Security analysis of full domain hash . . . . . . . . . . . . . . . 13.4.1 Repeated oneway functions: a useful lemma . . . . . . 13.4.2 Proofs of Theorems 13.3 and 13.4 . . . . . . . . . . . . . 13.5 An RSAbased signature scheme with tighter security proof . . 13.6 Case study: PKCS1 signatures . . . . . . . . . . . . . . . . . . 13.6.1 Bleichenbacher’s attack on PKCS1 signatures . . . . . . 13.7 Signcryption: combining signatures and encryption . . . . . . . 13.7.1 Secure signcryption . . . . . . . . . . . . . . . . . . . . . 13.7.2 Signcryption as an abstract interface . . . . . . . . . . . 13.7.3 Constructions: encryptthensign and signthenencrypt 13.7.4 A construction based on DiffieHellman key exchange . 13.7.5 Additional desirable properties . . . . . . . . . . . . . .
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497 499 500 503 503 504 505 506 509 509 513 514 516 518 519 521 523 526 530 532
13.8 Certificates and the publickey infrastructure . . . . . . . . . . . 13.8.1 Coping with malicious or negligent certificate authorities . 13.8.2 Certificate revocation . . . . . . . . . . . . . . . . . . . . 13.9 Case study: legal aspects of digital signatures . . . . . . . . . . . 13.10A fun application: private information retrieval . . . . . . . . . . 13.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14 Fast signatures from oneway functions 14.1 Lamport signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 A general Lamport framework . . . . . . . . . . . . . . . . 14.1.2 Optimized Lamport . . . . . . . . . . . . . . . . . . . . . . 14.2 HORS signatures: Lamport in the random oracle model . . . . . . 14.2.1 MerkleHORS: reducing the public key size . . . . . . . . . 14.3 Comparing onetime signatures . . . . . . . . . . . . . . . . . . . . 14.4 Applications of onetime signatures . . . . . . . . . . . . . . . . . . 14.4.1 Online/o✏ine signatures from onetime signatures . . . . . 14.4.2 Authenticating streamed data with onetime signatures . . 14.5 Merkle stateless signatures: manytime signatures from onetime signatures . . . . . . . . . . . 14.5.1 Extending the number of signatures from a qtime signature 14.5.2 The complete Merkle stateless signature system . . . . . . . 14.5.3 Stateful Merkle signatures . . . . . . . . . . . . . . . . . . . 14.5.4 Comparing Merkle constructions . . . . . . . . . . . . . . . 14.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Analysis of number theoretic assumptions 15.1 How reasonable are the factoring and RSA assumptions? 15.1.1 Quadratic resudousity assumption . . . . . . . . 15.2 How reasonable are the DL and CDH assumptions? . . . 15.2.1 The Baby step giant step algorithm . . . . . . . 15.2.2 The PohligHellman algorithm . . . . . . . . . . 15.2.3 Information leakage . . . . . . . . . . . . . . . . 15.3 Discrete log in Z⇤p . . . . . . . . . . . . . . . . . . . . . . 15.3.1 The number field sieve . . . . . . . . . . . . . . . 15.3.2 Discretelog records in Z⇤p . . . . . . . . . . . . . 15.4 How reasonable is decision DiffieHellman? . . . . . . . . 15.5 Quantum attacks on number theoretic problems . . . . . 15.6 Side channel and fault attacks . . . . . . . . . . . . . . . 15.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . 15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
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16 Elliptic curve cryptography and pairings 16.1 The group of points of an elliptic curve . . . 16.2 Pairings . . . . . . . . . . . . . . . . . . . . 16.3 Signature schemes from pairings . . . . . . 16.4 Advanced encryption schemes from pairings 16.4.1 Identity based encryption . . . . . . 16.4.2 Attribute based encryption . . . . .
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582 . 582 . 582 . 582 . 582 . 582 . 582
17 Lattice based cryptography 17.1 Integer lattices . . . . . . . . . . . . . . . . . . . 17.2 Hard problems on lattices . . . . . . . . . . . . . 17.2.1 The SIS problem . . . . . . . . . . . . . . 17.2.2 The learning with errors (LWE) problem 17.3 Signatures from lattice problems . . . . . . . . . 17.4 Publickey encryption using lattices . . . . . . . . 17.5 Fully homomorphic encryption . . . . . . . . . .
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III
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Protocols
583 583 583 583 583 583 583 583
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18 Identification protocols 18.1 Interactive protocols: general notions . . . . . . . . . . . . . . 18.1.1 Mathematical details . . . . . . . . . . . . . . . . . . . 18.2 ID protocols: definitions . . . . . . . . . . . . . . . . . . . . . 18.3 Password protocols: security against direct attacks . . . . . . 18.3.1 Weak passwords and dictionary attacks . . . . . . . . 18.3.2 Preventing dictionary attacks: salts, peppers, and slow 18.3.3 More password management issues . . . . . . . . . . . 18.3.4 Case study: UNIX and Windows passwords . . . . . . 18.4 One time passwords: security against eavesdropping . . . . . 18.4.1 The SecurID system . . . . . . . . . . . . . . . . . . . 18.4.2 The S/key system . . . . . . . . . . . . . . . . . . . . 18.5 Challengeresponse: security against active attacks . . . . . . 18.5.1 Challengeresponse protocols . . . . . . . . . . . . . . 18.5.2 Concurrent attacks versus sequential attacks . . . . . 18.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Signatures from identification protocols 19.1 Schnorr’s identification protocol . . . . . . . . . . . 19.2 Honest verifier zero knowledge and security against 19.3 The GuillouQuisquater identification protocol . . 19.4 From identification protocols to signatures . . . . . 19.4.1 ⌃protocols . . . . . . . . . . . . . . . . . . 19.4.2 Signature construction . . . . . . . . . . . . 19.4.3 The Schnorr signature scheme . . . . . . . . xii
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19.4.4 The GQ signature scheme . . . . . 19.5 Secure against active attacks: OR proofs . 19.6 Nonce misuse resistance . . . . . . . . . . 19.7 Okamoto’s identification protocol . . . . . 19.8 Case study: the digital signature standard 19.8.1 Comparing signature schemes . . . 19.9 Notes . . . . . . . . . . . . . . . . . . . . 19.10Chapter summary . . . . . . . . . . . . . 19.11Exercises . . . . . . . . . . . . . . . . . .
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20 Authenticated Key Exchange 20.1 Identification and AKE . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 An encryptionbased protocol . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Insecure variations . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Forward secrecy and an ephemeral encryptionbased protocol . . . . . 20.3.1 Insecure variations . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Security of protocol EBKE . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Security of protocol EEBKE . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Explicit key confirmation . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Identity protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8.1 Insecure variations . . . . . . . . . . . . . . . . . . . . . . . . . 20.9 Onesided authenticated key exchange . . . . . . . . . . . . . . . . . . 20.9.1 Onesided authenticated variants of protocols EBKE and EEBKE . 20.9.2 Realworld security: phishing attacks . . . . . . . . . . . . . . . 20.10Noninterative key exchange . . . . . . . . . . . . . . . . . . . . . . . . 20.11Zero round trip key exchange . . . . . . . . . . . . . . . . . . . . . . . 20.12Password authenticated key exchange . . . . . . . . . . . . . . . . . . 20.12.1 Protocol PAKE0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.2 Protocol PAKE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.3 Protocol PAKE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.4 Protocol PAKE+ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.5 Explicit key confirmation . . . . . . . . . . . . . . . . . . . . . 20.12.6 Generic protection against server compromise . . . . . . . . . . 20.12.7 Phishing again . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.13Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.13.1 SSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.13.2 IKE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.14A fun application: establishing Tor channels . . . . . . . . . . . . . . . 20.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.16Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.17Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
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. . . . . . . . .
626 627 631 632 635 635 635 635 635
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
636 . 638 . 639 . 641 . 647 . 647 . 649 . 653 . 657 . 658 . 659 . 660 . 662 . 663 . 664 . 665 . 667 . 667 . 667 . 668 . 669 . 671 . 673 . 675 . 675 . 675 . 676 . 676 . 676 . 676 . 676 . 676 . 676
21 Key establishment with online Trusted Third Parties 21.1 A key exchange protocol with an online TTP . . . . . . 21.2 Insecure variations of protocol OnlineTTP . . . . . . . . 21.3 Security proof for protocol OnlineTTP . . . . . . . . . . 21.4 Case study: Kerberos V5 . . . . . . . . . . . . . . . . . 21.5 O✏ine TTP vs. Online TTP . . . . . . . . . . . . . . . 21.6 A fun application: timespace tradeo↵s . . . . . . . . . . 21.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
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677 . 678 . 680 . 685 . 685 . 689 . 690 . 690 . 690
22 Twoparty and multiparty secure computation 691 22.1 Yao’s two party protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 22.2 Multiparty secure computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
IV
Appendices
A Basic number theory A.1 Cyclic groups . . . . . . . . . . . . . A.2 Arithmetic modulo primes . . . . . . A.2.1 Basic concepts . . . . . . . . A.2.2 Structure of Z⇤p . . . . . . . . A.2.3 Quadratic residues . . . . . . A.2.4 Computing in Zp . . . . . . . A.2.5 Summary: arithmetic modulo A.3 Arithmetic modulo composites . . .
692 . . . . . . . . . . . . . . . . . . . . . . . . primes . . . .
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B Basic probability theory B.1 Birthday Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 More collision bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 A simple distinguisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
693 693 693 693 694 694 695 695 696
698 . 698 . 700 . 700
C Basic complexity theory
702
D Probabilistic algorithms
703
xiv
Part I
Secret key cryptography
3
Chapter 2
Encryption Roughly speaking, encryption is the problem of how two parties can communicate in secret in the presence of an eavesdropper. The main goals of this chapter are to develop a meaningful and useful definition of what we are trying to achieve, and to take some first steps in actually achieving it.
2.1
Introduction
Suppose Alice and Bob share a secret key k, and Alice wants to transmit a message m to Bob over a network while maintaining the secrecy of m in the presence of an eavesdropping adversary. This chapter begins the development of basic techniques to solve this problem. Besides transmitting a message over a network, these same techniques allow Alice to store a file on a disk so that no one else with access to the disk can read the file, but Alice herself can read the file at a later time. We should stress that while the techniques we develop to solve this fundamental problem are important and interesting, they do not by themselves solve all problems related to “secure communication.” • The techniques only provide secrecy in the situation where Alice transmits a single message per key. If Alice wants to secretly transmit several messages using the same key, then she must use methods developed in Chapter 5. • The techniques do not provide any assurances of message integrity: if the attacker has the ability to modify the bits of the ciphertext while it travels from Alice to Bob, then Bob may not realize that this happened, and accept a message other than the one that Alice sent. We will discuss techniques for providing message integrity in Chapter 6. • The techniques do not provide a mechanism that allow Alice and Bob to come to share a secret key in the first place. Maybe they are able to do this using some secure network (or a physical, facetoface meeting) at some point in time, while the message is sent at some later time when Alice and Bob must communicate over an insecure network. However, with an appropriate infrastructure in place, there are also protocols that allow Alice and Bob to exchange a secret key even over an insecure network: such protocols are discussed in Chapters 20 and 21.
4
2.2
Shannon ciphers and perfect security
2.2.1
Definition of a Shannon cipher
The basic mechanism for encrypting a message using a shared secret key is called a cipher (or encryption scheme). In this section, we introduce a slightly simplified notion of a cipher, which we call a Shannon cipher. A Shannon cipher is a pair E = (E, D) of functions. • The function E (the encryption function) takes as input a key k and a message m (also called a plaintext), and produces as output a ciphertext c. That is, c = E(k, m), and we say that c is the encryption of m under k. • The function D (the decryption function) takes as input a key k and a ciphertext c, and produces a message m. That is, m = D(k, c), and we say that m is the decryption of c under k. • We require that decryption “undoes” encryption; that is, the cipher must satisfy the following correctness property: for all keys k and all messages m, we have D(k, E(k, m) ) = m. To be slightly more formal, let us assume that K is the set of all keys (the key space), M is the set of all messages (the message space), and that C is the set of all ciphertexts (the ciphertext space). With this notation, we can write: E : K ⇥ M ! C,
D : K ⇥ C ! M. Also, we shall say that E is defined over (K, M, C). Suppose Alice and Bob want to use such a cipher so that Alice can send a message to Bob. The idea is that Alice and Bob must somehow agree in advance on a key k 2 K. Assuming this is done, then when Alice wants to send a message m 2 M to Bob, she encrypts m under k, obtaining the ciphertext c = E(k, m) 2 C, and then sends c to Bob via some communication network. Upon receiving c, Bob decrypts c under k, and the correctness property ensures that D(k, c) is the same as Alice’s original message m. For this to work, we have to assume that c is not tampered with in transit from Alice to Bob. Of course, the goal, intuitively, is that an eavesdropper, who may obtain c while it is in transit, does not learn too much about Alice’s message m — this intuitive notion is what the formal definition of security, which we explore below, will capture. In practice, keys, messages, and ciphertexts are often sequences of bytes. Keys are usually of some fixed length; for example, 16byte (i.e., 128bit) keys are very common. Messages and ciphertexts may be sequences of bytes of some fixed length, or of variable length. For example, a message may be a 1GB video file, a 10MB music file, a 1KB email message, or even a single bit encoding a “yes” or “no” vote in an electronic election. 5
Keys, messages, and ciphertexts may also be other types of mathematical objects, such as integers, or tuples of integers (perhaps lying in some specified interval), or other, more sophisticated types of mathematical objects (polynomials, matrices, or group elements). Regardless of how fancy these mathematical objects are, in practice, they must at some point be represented as sequences of bytes for purposes of storage in, and transmission between, computers. For simplicity, in our mathematical treatment of ciphers, we shall assume that K, M, and C are sets of finite size. While this simplifies the theory, it means that if a realworld system allows messages of unbounded length, we will (somewhat artificially) impose a (large) upper bound on legal message lengths. To exercise the above terminology, we take another look at some of the example ciphers discussed in Chapter 1. Example 2.1. A onetime pad is a Shannon cipher E = (E, D), where the keys, messages, and ciphertexts are bit strings of the same length; that is, E is defined over (K, M, C), where K := M := C := {0, 1}L , for some fixed parameter L. For a key k 2 {0, 1}L and a message m 2 {0, 1}L the encryption function is defined as follows: E(k, m) := k m, and for a key k 2 {0, 1}L and ciphertext c 2 {0, 1}L , the decryption function is defined as follows: D(k, c) := k
c.
Here, “ ” denotes bitwise exclusiveOR, or in other words, componentwise addition modulo 2, and satisfies the following algebraic laws: for all bit vectors x, y, z 2 {0, 1}L , we have x
y=y
x,
x
(y
z) = (x
y)
z,
x
0L = x,
and
x
x = 0L .
These properties follow immediately from the corresponding properties for addition modulo 2. Using these properties, it is easy to check that the correctness property holds for E: for all k, m 2 {0, 1}L , we have D(k, E(k, m) ) = D(k, k
m) = k
(k
m) = (k
k)
m = 0L
m = m.
The encryption and decryption functions happen to be the same in this case, but of course, not all ciphers have this property. 2 Example 2.2. A variable length onetime pad is a Shannon cipher E = (E, D), where the keys are bit strings of some fixed length L, while messages and ciphertexts are variable length bit strings, of length at most L. Thus, E is defined over (K, M, C), where K := {0, 1}L
and
M := C := {0, 1}L .
for some parameter L. Here, {0, 1}L denotes the set of all bit strings of length at most L (including the empty string). For a key k 2 {0, 1}L and a message m 2 {0, 1}L of length `, the encryption function is defined as follows: E(k, m) := k[0 . . ` 1] m, 6
and for a key k 2 {0, 1}L and ciphertext c 2 {0, 1}L of length `, the decryption function is defined as follows: D(k, c) := k[0 . . ` 1] c. Here, k[0 . . ` 1] denotes the truncation of k to its first ` bits. The reader may verify that the correctness property holds for E. 2 Example 2.3. A substitution cipher is a Shannon cipher E = (E, D) of the following form. Let ⌃ be a finite alphabet of symbols (e.g., the letters A–Z, plus a space symbol, ). The message space M and the ciphertext space C are both sequences of symbols from ⌃ of some fixed length L: M := C := ⌃L . The key space K consists of all permutations on ⌃; that is, each k 2 K is a onetoone function from ⌃ onto itself. Note that K is a very large set; indeed, K = ⌃! (for ⌃ = 27, K ⇡ 1.09 · 1028 ). Encryption of a message m 2 ⌃L under a key k 2 K (a permutation on ⌃) is defined as follows E(k, m) :=
k(m[0]), k(m[1]), . . . , k(m[L
1]) ,
where m[i] denotes the ith entry of m (counting from zero), and k(m[i]) denotes the application of the permutation k to the symbol m[i]. Thus, to encrypt m under k, we simply apply the permutation k componentwise to the sequence m. Decryption of a ciphertext c 2 ⌃L under a key k 2 K is defined as follows: D(k, c) :=
k
1
(c[0]), k
1
(c[1]), . . . , k
1
(c[L
1]) .
Here, k 1 is the inverse permutation of k, and to decrypt c under k, we simply apply k 1 componentwise to the sequence c. The correctness property is easily verified: for a message m 2 ⌃L and key k 2 K, we have D(k, E(k, m) ) = D(k, (k(m[0]), k(m[1]), . . . , k(m[L = (k
1
(k(m[0])), k
1
(k(m[1])), . . . , k
= (m[0], m[1], . . . , m[L
1]) = m.
1]) ) 1
(k(m[L
1])))
2
Example 2.4 (additive onetime pad). We may also define a “addition mod n” variation of the onetime pad. This is a cipher E = (E, D), defined over (K, M, C), where K := M := C := {0, . . . , n 1}, where n is a positive integer. Encryption and decryption are defined as follows: E(k, m) := m + k mod n
D(k, c) := c
k mod n.
The reader may easily verify that the correctness property holds for E. 2
2.2.2
Perfect security
So far, we have just defined the basic syntax and correctness requirements of a Shannon cipher. Next, we address the question: what is a “secure” cipher? Intuitively, the answer is that a secure cipher is one for which an encrypted message remains “well hidden,” even after seeing its encryption. However, turning this intuitive answer into one that is both mathematically meaningful and practically relevant is a real challenge. Indeed, although ciphers have been used for centuries, it 7
is only in the last few decades that mathematically acceptable definitions of security have been developed. In this section, we develop the mathematical notion of perfect security — this is the “gold standard” for security (at least, when we are only worried about encrypting a single message and do not care about integrity). We will also see that it is possible to achieve this level of security; indeed, we will show that the onetime pad satisfies the definition. However, the onetime pad is not very practical, in the sense that the keys must be as long as the messages: if Alice wants to send a 1GB file to Bob, they must already share a 1GB key! Unfortunately, this cannot be avoided: we will also prove that any perfectly secure cipher must have a key space at least as large as its message space. This fact provides the motivation for developing a definition of security that is weaker, but that is acceptable from a practical point of view, and which allows one to encrypt long messages using short keys. If Alice encrypts a message m under a key k, and an eavesdropping adversary obtains the ciphertext c, Alice only has a hope of keeping m secret if the key k is hard to guess, and that means, at the very least, that the key k should be chosen at random from a large key space. To say that m is “well hidden” must at least mean that it is hard to completely determine m from c, without knowledge of k; however, this is not really enough. Even though the adversary may not know k, we assume that he does know the encryption algorithm and the distribution of k. In fact, we will assume that when a message is encrypted, the key k is always chosen at random, uniformly from among all keys in the key space. The adversary may also have some knowledge of the message encrypted — because of circumstances, he may know that the set of possible messages is quite small, and he may know something about how likely each possible message is. For example, suppose he knows the message m is either m0 = "ATTACK AT DAWN" or m1 = "ATTACK AT DUSK", and that based on the adversary’s available intelligence, Alice is equally likely to choose either one of these two messages. This, without seeing the ciphertext c, the adversary would only have a 50% chance of guessing which message Alice sent. But we are assuming the adversary does know c. Even with this knowledge, both messages may be possible; that is, there may exist keys k0 and k1 such that E(k0 , m0 ) = c and E(k1 , m1 ) = c, so he cannot be sure if m = m0 or m = m1 . However, he can still guess. Perhaps it is a property of the cipher that there are 800 keys k0 such that E(k0 , m0 ) = c, and 600 keys k1 such that E(k1 , m1 ) = c. If that is the case, the adversary’s best guess would be that m = m0 . Indeed, the probability that this guess is correct is equal to 800/(800 + 600) ⇡ 57%, which is better than the 50% chance he would have without knowledge of the ciphertext. Our formal definition of perfect security expressly rules out the possibility that knowledge of the ciphertext increases the probability of guessing the encrypted message, or for that matter, determining any property of the message whatsoever. Without further ado, we formally define perfect security. In this definition, we will consider a probabilistic experiment in which the key is drawn uniformly from the key space. We write k to denote the random variable representing this random key. For a message m, E(k, m) is another random variable, which represents the application of the encryption function to our random key and the message m. Thus, every message m gives rise to a di↵erent random variable E(k, m). Definition 2.1 (perfect security). Let E = (E, D) be a Shannon cipher defined over (K, M, C). Consider a probabilistic experiment in which the random variable k is uniformly distributed over K. If for all m0 , m1 2 M, and all c 2 C, we have Pr[E(k, m0 ) = c] = Pr[E(k, m1 ) = c], 8
then we say that E is a perfectly secure Shannon cipher. There are a number of equivalent formulations of perfect security that we shall explore. We state a couple of these here. Theorem 2.1. Let E = (E, D) be a Shannon cipher defined over (K, M, C). The following are equivalent: (i) E is perfectly secure. (ii) For every c 2 C, there exists Nc (possibly depending on c) such that for all m 2 M, we have {k 2 K : E(k, m) = c} = Nc . (iii) If the random variable k is uniformly distributed over K, then each of the random variables E(k, m), for m 2 M, has the same distribution. Proof. To begin with, let us restate (ii) as follows: for every c 2 C, there exists a number Pc (depending on c) such that for all m 2 M, we have Pr[E(k, m) = c] = Pc . Here, k is a random variable uniformly distributed over K. Note that Pc = Nc /K, where Nc is as in the original statement of (ii). This version of (ii) is clearly the same as (iii). (i) =) (ii). We prove (ii) assuming (i). To prove (ii), let c 2 C be some fixed ciphertext. Pick some arbitrary message m0 2 M, and let Pc := Pr[E(k, m0 ) = c]. By (i), we know that for all m 2 M, we have Pr[E(k, m) = c] = Pr[E(k, m0 ) = c] = Pc . That proves (ii). (ii) =) (i). We prove (i) assuming (ii). Consider any fixed m0 , m1 2 M and c 2 C. (ii) says that Pr[E(k, m0 ) = c] = Pc = Pr[E(k, m1 ) = c], which proves (i). 2 As promised, we give a proof that the onetime pad (see Example 2.1) is perfectly secure. Theorem 2.2. The onetime pad is a perfectly secure Shannon cipher. Proof. Suppose that the Shannon cipher E = (E, D) is a onetime pad, and is defined over (K, M, C), where K := M := C := {0, 1}L . For any fixed message m 2 {0, 1}L and ciphertext c 2 {0, 1}L , there is a unique key k 2 {0, 1}L satisfying the equation k namely, k := m 2
m = c,
c. Therefore, E satisfies condition (ii) in Theorem 2.1 (with Nc = 1 for each c).
Example 2.5. Consider again the variable length onetime pad, defined in Example 2.2. This does not satisfy our definition of perfect security, since a ciphertext has the same length as the corresponding plaintext. Indeed, let us choose an arbitrary string of length 1, call it m0 , and an arbitrary string of length 2, call it m1 . In addition, suppose that c is an arbitrary length 1 string, and that k is a random variable that is uniformly distributed over the key space. Then we have Pr[E(k, m0 ) = c] = 1/2
and
Pr[E(k, m1 ) = c] = 0,
which provides a direct counterexample to Definition 2.1. 9
Intuitively, the variable length onetime pad cannot satisfy our definition of perfect security simply because any ciphertext leaks the length of the corresponding plaintext. However, in some sense (which we do not make precise right now), this is the only information leaked. It is perhaps not clear whether this should be viewed as a problem with the cipher or with our definition of perfect security. On the one hand, one can imagine scenarios where the length of a message may vary greatly, and while we could always “pad” short messages to e↵ectively make all messages equally long, this may be unacceptable from a practical point of view, as it is a waste of bandwidth. On the other hand, one must be aware of the fact that in certain applications, leaking just the length of a message may be dangerous: if you are encrypting a “yes” or “no” answer to a question, just the length of the obvious ASCII encoding of these strings leaks everything, so you better pad “no” out to three characters. 2 Example 2.6. Consider again the substitution cipher defined in Example 2.3. There are a couple of di↵erent ways to see that this cipher is not perfectly secure. For example, choose a pair of messages m0 , m1 2 ⌃L such that the first two components of m0 are equal, yet the first two components of m1 are not equal; that is, m0 [0] = m0 [1]
and
m1 [0] 6= m1 [1].
Then for each key k, which is a permutation on ⌃, if c = E(k, m0 ), then c[0] = c[1], while if c = E(k, m1 ), then c[0] 6= c[1]. In particular, it follows that if k is uniformly distributed over the key space, then the distributions of E(k, m0 ) and E(k, m1 ) will not be the same. Even the weakness described in the previous paragraph may seem somewhat artificial. Another, perhaps more realistic, type of attack on the substitution cipher works as follows. Suppose the substitution cipher is used to encrypt email messages. As anyone knows, an email starts with a “standard header,” such as "FROM". Suppose the ciphertext is c 2 ⌃L is intercepted by an adversary. The secret key is actually a permutation k on ⌃. The adversary knows that c[0 . . . 3] = (k(F), k(R), k(O), k(M)). Thus, if the original message is m 2 ⌃L , the adversary can now locate all positions in m where an F occurs, where an R occurs, where an O occurs, and where an M occurs. Based just on this information, along with specific, contextual information about the message, together with general information about letter frequencies, the adversary may be able to deduce quite a bit about the original message. 2 Example 2.7. Consider the additive onetime pad, defined in Example 2.4. It is easy to verity that this is perfectly secure. Indeed, it satisfies condition (ii) in Theorem 2.1 (with Nc = 1 for each c). 2 The next two theorems develop two more alternative characterizations of perfect security. For the first, suppose an eavesdropping adversary applies some predicate to a ciphertext he has obtained. The predicate (which is a booleanvalued function on the ciphertext space) may be something very simple, like the parity function (i.e., whether the number of 1 bits in the ciphertext is even or odd), or it might be some more elaborate type of statistical test. Regardless of how clever or complicated the predicate is, perfect security guarantees that the value of this predicate on the ciphertext reveals nothing about the message.
10
Theorem 2.3. Let E = (E, D) be a Shannon cipher defined over (K, M, C). Consider a probabilistic experiment in which k is a random variable uniformly distributed over K. Then E is perfectly secure if and only if for every predicate on C, for all m0 , m1 2 M, we have Pr[ (E(k, m0 ))] = Pr[ (E(k, m1 ))]. Proof. This is really just a simple calculation. On the one hand, suppose E is perfectly secure, and let , m0 , and m1 be given. Let S := {c 2 C : (c)}. Then we have X X Pr[E(k, m0 ) = c] = Pr[E(k, m1 ) = c] = Pr[ (E(k, m1 ))]. Pr[ (E(k, m0 ))] = c2S
c2S
Here, we use the assumption that E is perfectly secure in establishing the second equality. On the other hand, suppose E is not perfectly secure, so there exist m0 , m1 , and c such that Pr[E(k, m0 ) = c] 6= Pr[E(k, m1 ) = c]. Defining to be the predicate that is true for this particular c, and false for all other ciphertexts, we see that Pr[ (E(k, m0 ))] = Pr[E(k, m0 ) = c] 6= Pr[E(k, m1 ) = c] = Pr[ (E(k, m1 ))].
2
The next theorem states in yet another way that perfect security guarantees that the ciphertext reveals nothing about the message. Suppose that m is a random variable distributed over the message space M. We do not assume that m is uniformly distributed over M. Now suppose k is a random variable uniformly distributed over the key space K, independently of m, and define c := E(k, m), which is a random variable distributed over the ciphertext space C. The following theorem says that perfect security guarantees that c and m are independent random variables. One way of characterizing this independence is to say that for each ciphertext c 2 C that occurs with nonzero probability, and each message m 2 M, we have Pr[m = m  c = c] = Pr[m = m]. Intuitively, this means that after seeing a ciphertext, we have no more information about the message than we did before seeing the ciphertext. Another way of characterizing this independence is to say that for each message m 2 M that occurs with nonzero probability, and each ciphertext c 2 C, we have Pr[c = c  m = m] = Pr[c = c]. Intuitively, this means that the choice of message has no impact on the distribution of the ciphertext. The restriction that m and k are independent random variables is sensible: in using any cipher, it is a very bad idea to choose the key in a way that depends on the message, or vice versa (see Exercise 2.16). Theorem 2.4. Let E = (E, D) be a Shannon cipher defined over (K, M, C). Consider a random experiment in which k and m are random variables, such that • k is uniformly distributed over K, 11
• m is distributed over M, and • k and m are independent. Define the random variable c := E(k, m). Then we have: • if E is perfectly secure, then c and m are independent; • conversely, if c and m are independent, and each message in M occurs with nonzero probability, then E is perfectly secure. Proof. We define M⇤ to be the set of messages that occur with nonzero probability. We begin with a simple observation. Consider any fixed m 2 M⇤ and c 2 C. Then we have Pr[c = c  m = m] = Pr[E(k, m) = c  m = m], and since k and m are independent, so are E(k, m) and m, and hence Pr[E(k, m) = c  m = m] = Pr[E(k, m) = c]. Putting this all together, we have: Pr[c = c  m = m] = Pr[E(k, m) = c].
(2.1)
We now prove the first implication. So assume that E is perfectly secure. We want to show that c and m are independent. To to this, let m 2 M⇤ and c 2 C be given. It will suffice to show that Pr[c = c  m = m] = Pr[c = c]. We have Pr[c = c] =
X
m0 2M⇤
=
X
Pr[c = c  m = m0 ] Pr[m = m0 ]
(by total probability)
Pr[E(k, m0 ) = c] Pr[m = m0 ]
(by (2.1))
m0 2M⇤
=
X
Pr[E(k, m) = c] Pr[m = m0 ]
m0 2M
= Pr[E(k, m) = c] = Pr[E(k, m) = c]
X
(by the definition of perfect security)
Pr[m = m0 ]
m0 2M⇤
= Pr[c = c  m = m]
(probabilities sum to 1) (again by (2.1))
This shows that c and m are independent. That proves the first implication. For the second, we assume that c and m are independent, and moreover, that every message occurs with nonzero probability (so M⇤ = M). We want to show that E is perfectly secure, which means that for each m0 , m1 2 M, and each c 2 C, we have Pr[E(k, m0 ) = c] = Pr[E(k, m1 ) = c]. 12
(2.2)
But we have Pr[E(k, m0 ) = c] = Pr[c = c  m = m0 ] = Pr[c = c]
(by (2.1))
(by independence of c and m)
= Pr[c = c  m = m1 ] = Pr[E(k, m1 ) = c]
(again by independence of c and m) (again by (2.1)).
That shows that E is perfectly secure. 2
2.2.3
The bad news
We have saved the bad news for last. The next theorem shows that perfect security is such a powerful notion that one can really do no better than the onetime pad: keys must be at least as long as messages. As a result, it is almost impossible to use perfectly secure ciphers in practice: if Alice wants to send Bob a 1GB video file, then Alice and Bob have to agree on a 1GB secret key in advance. Theorem 2.5 (Shannon’s theorem). Let E = (E, D) be a Shannon cipher defined over (K, M, C). If E is perfectly secure, then K M. Proof. Assume that K < M. We want to show that E is not perfectly secure. To this end, we show that there exist messages m0 and m1 , and a ciphertext c, such that Pr[E(k, m0 ) = c] > 0, and
(2.3)
Pr[E(k, m1 ) = c] = 0.
(2.4)
Here, k is a random variable, uniformly distributed over K. To do this, choose any message m0 2 M, and any key k0 2 K. Let c := E(k0 , m0 ). It is clear that (2.3) holds. Next, let S := {D(k1 , c) : k1 2 K}. Clearly, S K < M, and so we can choose a message m1 2 M \ S. To prove (2.4), we need to show that there is no key k1 such that E(k1 , m1 ) = c. Assume to the contrary that E(k1 , m1 ) = c for some k1 ; then for this key k1 , by the correctness property for ciphers, we would have D(k1 , c) = D(k1 , E(k1 , m1 ) ) = m1 , which would imply that m1 belongs to S, which is not the case. That proves (2.4), and the theorem follows. 2
2.3
Computational ciphers and semantic security
As we have seen in Shannon’s theorem (Theorem 2.5), the only way to achieve perfect security is to have keys that are as long as messages. However, this is quite impractical: we would like to be 13
able to encrypt a long message (say, a document of several megabytes) using a short key (say, a few hundred bits). The only way around Shannon’s theorem is to relax our security requirements. The way we shall do this is to consider not all possible adversaries, but only computationally feasible adversaries, that is, “real world” adversaries that must perform their calculations on real computers using a reasonable amount of time and memory. This will lead to a weaker definition of security called semantic security. Furthermore, our definition of security will be flexible enough to allow ciphers with variable length message spaces to be considered secure so long as they do not leak any useful information about an encrypted message to an adversary other than the length of message. Also, since our focus is now on the “practical,” instead of the “mathematically possible,” we shall also insist that the encryption and decryption functions are themselves efficient algorithms, and not just arbitrary functions.
2.3.1
Definition of a computational cipher
A computational cipher E = (E, D) is a pair of efficient algorithms, E and D. The encryption algorithm E takes as input a key k, along with a message m, and produces as output a ciphertext c. The decryption algorithm D takes as input a key k, a ciphertext c, and outputs a message m. Keys lie in some finite key space K, messages lie in a finite message space M, and ciphertexts lie in some finite ciphertext space C. Just as for a Shannon cipher, we say that E is defined over (K, M, C). Although it is not really necessary for our purposes in this chapter, we will allow the encryption function E to be a probabilistic algorithm (see Chapter D). This means that for fixed inputs k and m, the output of E(k, m) may be one of many values. To emphasize the probabilistic nature of this computation, we write c R E(k, m) to denote the process of executing E(k, m) and assigning the output to the program variable c. We shall use this notation throughout the text whenever we use probabilistic algorithms. Similarly, we write k R K to denote the process of assigning to the program variable k a random, uniformly distributed element of from the key space K. We shall use the analogous notation to sample uniformly from any finite set. We will not see any examples of probabilistic encryption algorithms in this chapter (we will see our first examples of this in Chapter 5). Although one could allow the decryption algorithm to be probabilistic, we will have no need for this, and so will only discuss ciphers with deterministic decryption algorithms. However, it will be occasionally be convenient to allow the decryption algorithm to return a special reject value (distinct from all messages), indicating some kind of error occurred during the decryption process. Since the encryption algorithm is probabilistic, for a given key k and message m, the encryption algorithm may output one of many possible ciphertexts; however, each of these possible ciphertexts should decrypt to m. We can state this correctness requirement more formally as follows: for all keys k 2 K and messages m 2 M, if we execute c
R
E(k, m), m0
D(k, c),
then m = m0 with probability 1.
14
From now on, whenever we refer to a cipher, we shall mean a computational cipher, as defined above. Moreover, if the encryption algorithm happens to be deterministic, then we may call the cipher a deterministic cipher. Observe that any deterministic cipher is a Shannon cipher; however, a computational cipher need not be a Shannon cipher (if it has a probabilistic encryption algorithm), and a Shannon cipher need not be a computational cipher (if its encryption or decryption operations have no efficient implementations). Example 2.8. The onetime pad (see Example 2.1) and the variable length onetime pad (see Example 2.2) are both deterministic ciphers, since their encryption and decryption operations may be trivially implemented as efficient, deterministic algorithms. The same holds for the substitution cipher (see Example 2.3), provided the alphabet ⌃ is not too large. Indeed, in the obvious implementation, a key — which is a permutation on ⌃ — will be represented by an array indexed by ⌃, and so we will require O(⌃) space just to store a key. This will only be practical for reasonably sized ⌃. The additive onetime pad discussed in Example 2.4 is also a deterministic cipher, since both encryption and decryption operations may be efficiently implemented (if n is large, special software to do arithmetic with large integers may be necessary). 2
2.3.2
Definition of semantic security
To motivate the definition of semantic security, consider a deterministic cipher E = (E, D), defined over (K, M, C). Consider again the formulation of perfect security in Theorem 2.3. This says that for all predicates on the ciphertext space, and all messages m0 , m1 , we have Pr[ (E(k, m0 ))] = Pr[ (E(k, m1 ))],
(2.5)
where k is a random variable uniformly distributed over the key space K. Instead of insisting that these probabilities are equal, we shall only require that they are very close; that is, Pr[ (E(k, m0 ))]
Pr[ (E(k, m1 ))] ✏,
(2.6)
for some very small, or negligible, value of ✏. By itself, this relaxation does not help very much (see Exercise 2.5). However, instead of requiring that (2.6) holds for every possible , m0 , and m1 , we only require that (2.6) holds for all messages m0 and m1 that can be generated by some efficient algorithm, and all predicates that can be computed by some efficient algorithm (these algorithms could be probabilistic). For example, suppose it were the case that using the best possible algorithms for generating m0 and m1 , and for testing some predicate , and using (say) 10,000 computers in parallel for 10 years to perform these calculations, (2.6) holds for ✏ = 2 100 . While not perfectly secure, we might be willing to say that the cipher is secure for all practical purposes. Also, in defining semantic security, we address an issue raised in Example 2.5. In that example, we saw that the variable length onetime pad did not satisfy the definition of perfect security. However, we want our definition to be flexible enough so that ciphers like the variable length onetime pad, which e↵ectively leak no information about an encrypted message other than its length, may be considered secure as well. Now the details. To precisely formulate the definition of semantic security, we shall describe an attack game played between two parties: the challenger and an adversary. As we will see, the 15
Challenger
m0 , m 1 2 M
(Experiment b) k
R
c
R
K
A
c
E(k, mb )
ˆb 2 {0, 1}
Figure 2.1: Experiment b of Attack Game 2.1 challenger follows a very simple, fixed protocol. However, an adversary A may follow an arbitrary (but still efficient) protocol. The challenger and the adversary A send messages back and forth to each other, as specified by their protocols, and at the end of the game, A outputs some value. Actually, our attack game for defining semantic security comprises two alternative “subgames,” or “experiments” — in both experiments, the adversary follows the same protocol; however, the challenger’s behavior is slightly di↵erent in the two experiments. The attack game also defines a probability space, and this in turn defines the adversary’s advantage, which measures the di↵erence between the probabilities of two events in this probability space. Attack Game 2.1 (semantic security). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define Experiment b: • The adversary computes m0 , m1 2 M, of the same length, and sends them to the challenger. • The challenger computes k
R
K, c
R
E(k, mb ), and sends c to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s semantic security advantage with respect to E as SSadv[A, E] := Pr[W0 ]
Pr[W1 ] .
2
Note that in the above game, the events W0 and W1 are defined with respect to the probability space determined by the random choice of k, the random choices made (if any) by the encryption algorithm, and the random choices made (if any) by the adversary. The value SSadv[A, E] is a number between 0 and 1. See Fig. 2.1 for a schematic diagram of Attack Game 2.1. As indicated in the diagram, A’s “output” is really just a final message to the challenger. 16
Definition 2.2 (semantic security). A cipher E is semantically secure if for all efficient adversaries A, the value SSadv[A, E] is negligible. As a formal definition, this is not quite complete, as we have yet to define what we mean by “messages of the same length”, “efficient adversaries”, and “negligible”. We will come back to this shortly. Let us relate this formal definition to the discussion preceding it. Suppose that the adversary A in Attack Game 2.1 is deterministic. First, the adversary computes in a deterministic fashion messages m0 , m1 , and then evaluates a predicate on the ciphertext c, outputting 1 if true and 0 if false. Semantic security says that the value ✏ in (2.6) is negligible. In the case where A is probabilistic, we can view A as being structured as follows: it generates a random value r from (r) (r) some appropriate set, and deterministically computes messages m0 , m1 , which depend on r, and evaluates a predicate (r) on c, which also depends on r. Here, semantic security says that the value (r) (r) ✏ in (2.6), with m0 , m1 , replaced by m0 , m1 , (r) , is negligible — but where now the probability is with respect to a randomly chosen key and a randomly chosen value of r. Remark 2.1. Let us now say a few words about the requirement that the messages m0 and m1 computed by the adversary Attack Game 2.1 be of the same length. • First, the notion of the “length” of a message is specific to the particular message space M; in other words, in specifying a message space, one must specify a rule that associates a length (which is a nonnegative integer) with any given message. For most concrete message spaces, this will be clear: for example, for the message space {0, 1}L (as in Example 2.2), the length of a message m 2 {0, 1}L is simply its length, m, as a bit string. However, to make our definition somewhat general, we leave the notion of length as an abstraction. Indeed, some message spaces may have no particular notion of length, in which case all messages may be viewed as having length 0. • Second, the requirement that m0 and m1 be of the same length means that the adversary is not deemed to have broken the system just because he can e↵ectively distinguish an encryption of a message of one length from an encryption of a message of a di↵erent length. This is how our formal definition captures the notion that an encryption of a message is allowed to leak the length of the message (but nothing else). We already discussed in Example 2.5 how in certain applications, leaking the just length of the message can be catastrophic. However, since there is no general solution to this problem, most realworld encryption schemes (for example, TLS) do not make any attempt at all to hide the length of the message. This can lead to real attacks. For example, Chen et al. [25] show that the lengths of encrypted messages can reveal considerable information about private data that a user supplies to a cloud application. They use an online tax filing system as their example, but other works show attacks of this type on many other systems. 2 Example 2.9. Let E be a deterministic cipher that is perfectly secure. Then it is easy to see that for every adversary A (efficient or not), we have SSadv[A, E] = 0. This follows almost immediately from Theorem 2.3 (the only slight complication is that our adversary A in Attack Game 2.1 may be probabilistic, but this is easily dealt with). In particular, E is semantically secure. Thus, if E is the onetime pad (see Example 2.1), we have SSadv[A, E] = 0 for all adversaries A; in particular, the onetime pad is semantically secure. Because the definition of semantic security is a bit more 17
forgiving with regard to variable length message spaces, it is also easy to see that if E is the variable length onetime pad (see Example 2.2), then SSadv[A, E] = 0 for all adversaries A; in particular, the variable length onetime pad is also semantically secure. 2 We need to say a few words about the terms “efficient” and “negligible”. Below in Section 2.4 we will fill in the remaining details (they are somewhat tedious, and not really very enlightening). Intuitively, negligible means so small as to be “zero for all practical purposes”: think of a number like 2 100 — if the probability that you spontaneously combust in the next year is 2 100 , then you would not worry about such an event occurring any more than you would an event that occurred with probability 0. Also, an efficient adversary is one that runs ins a“reasonable” amount time. We introduce two other terms: • A value N is called superpoly is 1/N is negligible. • A polybounded value which intuitively a reasonably sized number — in particular, we can say that the running time of any efficient adversary is a polybounded value. Fact 2.6. If ✏ and ✏0 are negligible values, and Q and Q0 are polybounded values, then: (i) ✏ + ✏0 is a negligible value, (ii) Q + Q0 and Q · Q0 are polybounded values, and (iii) Q · ✏ is a negligible value. For now, the reader can just take these facts as axioms. Instead of dwelling on these technical issues, we discuss an example that illustrates how one typically uses this definition in analyzing the security of a larger system that uses a semantically secure cipher.
2.3.3
Connections to weaker notions of security
Message recovery attacks Intuitively, in a message recovery attack, an adversary is given an encryption of a random message, and is able to recover the message from the ciphertext with probability significantly better than random guessing, that is, probability 1/M. Of course, any reasonable notion of security should rule out such an attack, and indeed, semantic security does. While this may seem intuitively obvious, we give a formal proof of this. One of our motivations for doing this is to illustrate in detail the notion of a security reduction, which is the main technique used to reason about the security of systems. Basically, the proof will argue that any efficient adversary A that can e↵ectively mount a message recovery attack on E can be used to build an efficient adversary B that breaks the semantic security of E; since semantic security implies that no such B exists, we may conclude that no such A exists. To formulate this proof in more detail, we need a formal definition of a message recovery attack. As before, this is done by giving attack game, which is a protocol between a challenger and an adversary. Attack Game 2.2 (message recovery). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, the attack game proceeds as follows: • The challenger computes m
R
M, k
R
K, c 18
R
E(k, m), and sends c to the adversary.
• The adversary outputs a message m ˆ 2 M. Let W be the event that m ˆ = m. We say that A wins the game in this case, and we define A’s message recovery advantage with respect to E as MRadv[A, E] := Pr[W ]
1/M .
2
Definition 2.3 (security against message recovery). A cipher E is secure against message recovery if for all efficient adversaries A, the value MRadv[A, E] is negligible. Theorem 2.7. Let E = (E, D) be a cipher defined over (K, M, C). If E is semantically secure then E is secure against message recovery. Proof. Assume that E is semantically secure. Our goal is to show that E is secure against message recovery. To prove that E is secure against message recovery, we have to show that every efficient adversary A has negligible advantage in Attack Game 2.2. To show this, we let an arbitrary but efficient adversary A be given, and our goal now is to show that A’s message recovery advantage, MRadv[A, E], is negligible. Let p denote the probability that A wins the message recovery game, so that MRadv[A, E] = p 1/M . We shall show how to construct an efficient adversary B whose semantic security advantage in Attack Game 2.1 is related to A’s message recovery advantage as follows: MRadv[A, E] SSadv[B, E].
(2.7)
Since B is efficient, and since we are assume E is semantically secure, the righthand side of (2.7) is negligible, and so we conclude that MRadv[A, E] is negligible. So all that remains to complete the proof is to show how to construct an efficient B that satisfies (2.7). The idea is to use A as a “black box” — we do not have to understand the inner workings of A as at all. Here is how B works. Adversary B generates two random messages, m0 and m1 , and sends these to its own SS challenger. This challenger sends B a ciphertext c, which B forwards to A, as if it were coming from A’s MR challenger. When A outputs a message m, ˆ our adversary B compares m0 to m, ˆ and outputs ˆb = 1 if m0 = m, ˆ and ˆb = 1 otherwise. That completes the description of B, which is illustrated in Fig. ??. Note that the running time of B is essentially the same as that of A. We now analyze the B’s SS advantage, and relate this to A’s MR advantage. For b = 0, 1, let pb be the probability that B outputs 1 if B’s SS challenger encrypts mb . So by definition SSadv[B, E] = p1 p0 . On the one hand, when c is an encryption of m0 , the probability p0 is precisely equal to A’s probability of winning the message recovery game, so p0 = p. On the other hand, when c is an encryption of m1 , the adversary A’s output is independent of m0 , and so p1 = 1/M. It follows that SSadv[B, E] = p1 p0  = 1/M p = MRadv[A, E]. This proves (2.7). In fact, equality holds in (2.7), but that is not essential to the proof. 2 19
The reader should make sure that he or she understands the logic of this proof, as this type of proof will be used over and over again throughout the book. We shall review the important parts of the proof here, and give another way of thinking about it. The core of the proof was establishing the following fact: for every efficient MR adversary A that attacks E as in Attack Game 2.2, there exists an efficient SS adversary B that attacks E as in Attack Game 2.1 such that MRadv[A, E] SSadv[B, E]. (2.8) We are trying to prove that if E is semantically secure, then E is secure against message recovery. In the above proof, we argued that if E is semantically secure, then the righthand side of (2.8) must be negligible, and hence so must the lefthand side; since this holds for all efficient A, we conclude that E is secure against message recovery.
Another way to approach the proof of the theorem is to prove the contrapositive: if E is not secure against message recovery, then E is not semantically secure. So, let us assume that E is not secure against message recovery. This means there exists an efficient adversary A whose message recovery advantage is nonnegligible. Using A we build an efficient adversary B that satisfies (2.8). By assumption, MRadv[A, E] is nonnegligible, and (2.8) implies that SSadv[B, E] is nonnegligible. From this, we conclude that E is not semantically secure. Said even more briefly: to prove that semantic security implies security against message recovery, we show how to turn an efficient adversary that breaks message recovery into an efficient adversary that breaks semantic security.
We also stress that the adversary B constructed in the proof just uses A as a “black box.” In fact, almost all of the constructions we shall see are of this type: B is essentially just a wrapper around A, consisting of some simple and efficient “interface layer” between B’s challenger and a single running instance of A. Ideally, we want the computational complexity of the interface layer to not depend on the computational complexity of A; however, some dependence is unavoidable: if an attack game allows A to make multiple queries to its challenger, the more queries A makes, the more work must be performed by the interface layer, but this work should just depend on the number of such queries and not on the running time of A. Thus, we will say adversary B is an elementary wrapper around adversary A when it can be structured as above, as an efficient interface interacting with A. The salient properties are: • If B is an elementary wrapper around A, and A is efficient, then B is efficient. • If C is an elementary wrapper around B and B is an elementary wrapper around A, then C is an elementary wrapper around A. These notions are formalized in Section 2.4 (but again, they are extremely tedious). Computing individual bits of a message If an encryption scheme is secure, not only should it be hard to recover the whole message, but it should be hard to compute any partial information about the message. We will not prove a completely general theorem here, but rather, consider a specific example. Suppose E = (E, D) is a cipher defined over (K, M, C), where M = {0, 1}L . For m 2 M, we define parity(m) to be 1 if the number of 1’s in m is odd, and 0 otherwise. Equivalently, parity(m) is the exclusiveOR of all the individual bits of m. 20
We will show that if E is semantically secure, then given an encryption c of a random message m, it is hard to predict parity(m). Now, since parity(m) is a single bit, any adversary can predict this value correctly with probability 1/2 just by random guessing. But what we want to show is that no efficient adversary can do significantly better than random guessing. As a warm up, suppose there were an efficient adversary A that could predict parity(m) with probability 1. This means that for every message m, every key k, and every encryption c of m, when we give A the ciphertext c, it outputs the parity of m. So we could use A to build an SS adversary B that works as follows. Our adversary chooses two messages, m0 and m1 , arbitrarily, but with parity(m0 ) = 0 and parity(m1 ) = 1. Then it hands these two messages to its own SS challenger, obtaining a ciphertext c, which it then forwards to it A. After receiving c, adversary A outputs a bit ˆb, and B outputs this same bit ˆb as its own output. It is easy to see that B’s SS advantage is precisely 1: when its SS challenger encrypts m0 , it always outputs 0, and when its SS challenger encrypts m1 , it always outputs 1. This shows that if E is semantically secure, there is no efficient adversary that can predict parity with probability 1. However, we can say even more: if E is semantically secure, there is no efficient adversary that can predict parity with probability significantly better than 1/2. To make this precise, we give an attack game: Attack Game 2.3 (parity prediction). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, the attack game proceeds as follows: • The challenger computes m
R
M, k
R
K, c
R
E(k, m), and sends c to the adversary.
• The adversary outputs ˆb 2 {0, 1}. Let W be the event that ˆb = parity(m). We define A’s message recovery advantage with respect to E as Parityadv[A, E] := Pr[W ] 1/2 . 2 Definition 2.4 (parity prediction). A cipher E is secure against parity prediction if for all efficient adversaries A, the value Parityadv[A, E] is negligible. Theorem 2.8. Let E = (E, D) be a cipher defined over (K, M, C), and M = {0, 1}L . If E is semantically secure, then E is secure against parity prediction. Proof. As in the proof of Theorem 2.7, we give a proof by reduction. In particular, we will show that for every parity prediction adversary A that attacks E as in Attack Game 2.3, there exists an SS adversary B that attacks E as in Attack Game 2.1, where B is an elementary wrapper around A, such that 1 Parityadv[A, E] = · SSadv[B, E]. 2 Let A be a parity prediction adversary that predicts parity with probability 1/2 + ✏, so Parityadv[A, E] = ✏. Here is how we construct our SS adversary B. Our adversary B generates a random message m0 , and sets m1 m0 (0L 1 k 1); that is, m1 is that same as m0 , except that the last bit is flipped. In particularly, m0 and m1 have opposite parity.
21
Our adversary B sends the pair m0 , m1 to its own SS challenger, receives a ciphertext c from that challenger, and forwards c to A. When A outputs a bit ˆb, our adversary B outputs 1 if ˆb = parity(m0 ), and outputs 0, otherwise. For b = 0, 1, let pb be the probability that B outputs 1 if B’s SS challenger encrypts mb . So by definition SSadv[B, E] = p1 p0 . We claim that p0 = 1/2 + ✏ and p1 = 1/2 ✏. This because regardless of whether m0 or m1 is encrypted, the distribution of mb is uniform over M, and so in case b = 0, our parity predictor A will output parity(m0 ) with probability 1/2 + ✏, and when b = 1, our parity predictor A with output parity(m1 ) with probability 1/2 + ✏, and so outputs parity(m0 ) with probability 1 (1/2 + ✏) = 1/2 ✏. Therefore, SSadv[B, E] = p1 p0  = 2✏ = 2 · Parityadv[A, E], which proves the theorem. 2 We have shown that if an adversary can e↵ectively predict the parity of a message, then it can be used to break semantic security. Conversely, it turns out that if an adversary can break semantic security, he can e↵ectively predict some predicate of the message (see Exercise 3.15).
2.3.4
Consequences of semantic security
In this section, we examine the consequences of semantic security in the context of a specific example, namely, electronic gambling. The specific details of the example are not so important, but the example illustrates how one typically uses the assumption of semantic security in applications. Consider the following extremely simplified version of roulette, which is a game between the house and a player. The player gives the house 1 dollar. He may place one of two kinds of bets: • “high or low,” or • “even or odd.” After placing his bet, the house chooses a random number r 2 {0, 1, . . . , 36}. The player wins if r 6= 0, and if • he bet “high” and r > 18, • he bet “low” and r 18, • he bet “even” and r is even, • he bet “odd” and r is odd. If the player wins, the house pays him 2 dollars (for a net win of 1 dollar), and if the player looses, the house pays nothing (for a net loss of 1 dollar). Clearly, the house has a small, but not insignificant advantage in this game: the probability that the player wins is 18/37 ⇡ 48.65%. Now suppose that this game is played over the Internet. Also, suppose that for various technical reasons, the house publishes an encryption of r before the player places his bet (perhaps to be decrypted by some regulatory agency that shares a key with the house). The player is free to analyze this encryption before placing his bet, and of course, by doing so, the player could conceivably 22
House r k
c
R R
R
{0, 1, . . . , 36} K
E(k, r)
A
c bet
outcome
W (r, bet) outcome
Figure 2.2: Internet roulette increase his chances of winning. However, if the cipher is any good, the player’s chances should not increase by much. Let us prove this, assuming r is encrypted using a semantically secure cipher E = (E, D), defined over (K, M, C), where M = {0, 1, . . . , 36} (we shall view all messages in M as having the same length in this example). Also, from now in, let us call the player A, to stress the adversarial nature of the player, and assume that A’s strategy can be modeled as an efficient algorithm. The game is illustrated in Fig. 2.2. Here, bet denotes one of “high,” “low,” “even,” “odd.” Player A sends bet to the house, who evaluates the function W (r, bet), which is 1 if bet is a winning bet with respect to r, and 0 otherwise. Let us define IRadv[A] := Pr[W (r, bet) = 1]
18/37 .
Our goal is to prove the following theorem. Theorem 2.9. If E is semantically secure, then for every efficient player A, the quantity IRadv[A] is negligible. As we did in Section 2.3.3, we prove this by reduction. More concretely, we shall show that for every player A, there exists an SS adversary B, where B is an elementary wrapper around A, such that IRadv[A] = SSadv[B, E]. (2.9) Thus, if there were an efficient player A with a nonnegligible advantage, we would obtain an efficient SS adversary B that breaks the semantic security of E, which we are assuming is impossible. Therefore, there is no such A. To motivate and analyze our new adversary B, consider an “idealized” version of Internet roulette, in which instead of publishing an encryption of the actual value r, the house instead publishes an encryption of a “dummy”value, say 0. The logic of the ideal Internet roulette game is illustrated in Fig. 2.3. Note, however, that in the ideal Internet roulette game, the house still uses the actual value of r to determine the outcome of the game. Let p0 be the probability that A wins at Internet roulette, and let p1 be the probability that A wins at ideal Internet roulette. 23
House r k
c
R R
R
A
{0, 1, . . . , 36} K
E(k, 0)
c bet
outcome
W (r, bet) outcome
Figure 2.3: ideal Internet roulette Our adversary B is designed to play in Attack Game 2.1 so that if ˆb denotes B’s output in that game, then we have: • if B is placed in Experiment 0, then Pr[ˆb = 1] = p0 ;
• if B is placed in Experiment 1, then Pr[ˆb = 1] = p1 . The logic of adversary B is illustrated in Fig. 2.4. It is clear by construction that B satisfies the properties claimed above, and so in particular, SSadv[B, E] = p1
p0 .
(2.10)
Now, consider the probability p1 that A wins at ideal Internet roulette. No matter how clever A’s strategy is, he wins with probability 18/37, since in this ideal Internet roulette game, the value of bet is computed from c, which is statistically independent of the value of r. That is, ideal Internet roulette is equivalent to physical roulette. Therefore, IRadv[A] = p1
p0 .
(2.11)
Combining (2.10) and (2.11), we obtain (2.9). The approach we have used to analyze Internet roulette is one that we will see again and again. The basic idea is to replace a system component by an idealized version of that component, and then analyze the behavior of this new, idealized version of the system. Another lesson to take away from the above example is that in reasoning about the security of a system, what we view as “the adversary” depends on what we are trying to do. In the above analysis, we cobbled together a new adversary B out of several components: one component was the original adversary A, while other components were scavenged from other parts of the system (the algorithm of “the house,” in this example). This will be very typical in our security analyses throughout this text. Intuitively, if we imagine a diagram of the system, at di↵erent points in the security analysis, we will draw a circle around di↵erent components of the system to identify what we consider to be “the adversary” at that point in the analysis. 24
B
Challenger
R
(Experiment b) m0 , m 1 k
R
c
R
K
r {0, 1, . . . , 36} m0 r m1 0
A
c
E(k, mb )
bet ˆb
ˆb
W (r, bet)
Figure 2.4: The SS adversary B in Attack Game 2.1
2.3.5
Bit guessing: an alternative characterization of semantic security
The example in Section 2.3.4 was a typical example of how one could use the definition of semantic security to analyze the security properties of a larger system that makes use of a semantically secure cipher. However, there is another characterization of semantic security that is typically more convenient to work with when one is trying to prove that a given cipher satisfies the definition. In this alternative characterization, we define a new attack game. The role played by the adversary is exactly the same as before. However, instead of having two di↵erent experiments, there is just a single experiment. In this bitguessing version of the attack game, the challenger chooses b 2 {0, 1} at random and runs Experiment b of Attack Game 2.1; it is the adversary’s goal to guess the bit b with probability significantly better than 1/2. Here are the details: Attack Game 2.4 (semantic security: bitguessing version). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, the attack game runs as follows: • The adversary computes m0 , m1 2 M, of the same length, and sends them to the challenger. • The challenger computes b
R
{0, 1}, k
R
K, c
R
E(k, mb ), and sends c to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. We say that A wins the game if ˆb = b. 2 Fig. 2.5 illustrates Attack Game 2.4. Note that in this game, the event that the A wins the game is defined with respect to the probability space determined by the random choice of b and k, the random choices made (if any) of the encryption algorithm, and the random choices made (if any) by the adversary. 25
A
Challenger b k
R
c
R
R
{0, 1} K
m0 , m 1 2 M
c
E(k, mb )
ˆb 2 {0, 1}
Figure 2.5: Attack Game 2.4 Of course, any adversary can win the game with probability 1/2, simply by ignoring c completely and choosing ˆb at random (or alternatively, always choosing ˆb to be 0, or always choosing it to be 1). What we are interested in is how much better than random guessing an adversary can do. If W denotes the event that the adversary wins the bitguessing version of the attack game, then we are interested in the quantity Pr[W ] 1/2, which we denote by SSadv⇤ [A, E]. Then we have: Theorem 2.10. For every cipher E and every adversary A, we have SSadv[A, E] = 2 · SSadv⇤ [A, E].
(2.12)
Proof. This is just a simple calculation. Let p0 be the probability that the adversary outputs 1 in Experiment 0 of Attack Game 2.1, and let p1 be the probability that the adversary outputs 1 in Experiment 1 of Attack Game 2.1. Now consider Attack Game 2.4. From now on, all events and probabilities are with respect to this game. If we condition on the event that b = 0, then in this conditional probability space, all of the other random choices made by the challenger and the adversary are distributed in exactly the same way as the corresponding values in Experiment 0 of Attack Game 2.1. Therefore, if ˆb is the output of the adversary in Attack Game 2.4, we have Pr[ˆb = 1  b = 0] = p0 . By a similar argument, we see that Pr[ˆb = 1  b = 1] = p1 . So we have Pr[ˆb = b] = Pr[ˆb = b  b = 0] Pr[b = 0] + Pr[ˆb = b  b = 1] Pr[b = 1] = Pr[ˆb = 0  b = 0] · 12 + Pr[ˆb = 1  b = 1] · 12 ⇣ ⌘ 1 ˆ ˆ = 2 1 Pr[b = 1  b = 0] + Pr[b = 1  b = 1] = 12 (1
p0 + p1 ).
26
Therefore, SSadv⇤ [A, E] = Pr[ˆb = b]
1 2
= 12 p1
p0  =
1 2
· SSadv[A, E].
That proves the theorem. 2 Just as it is convenient to refer SSadv[A, E] as A’s “SS advantage,” we shall refer to SSadv⇤ [A, E] as A’s “bitguessing SS advantage.” A generalization As it turns out, the above situation is quite generic. Although we do not need it in this chapter, for future reference we indicate here how the above situation generalizes. There will be a number of situations we shall encounter where some particular security property, call it “X,” for some of cryptographic system, call it “S,” can be defined in terms of an attack game involving two experiments, Experiment 0 and Experiment 1, where the adversary A’s protocol is the same in both experiments, while that of the challenger is di↵erent. For b = 0, 1, we define Wb to be the event that A outputs 1 in Experiment b, and we define Xadv[A, S] := Pr[W0 ]
Pr[W1 ]
to be A’s “X advantage.” Just as above, we can always define a “bitguessing” version of the attack game, in which the challenger chooses b 2 {0, 1} at random, and then runs Experiment b as its protocol. If W is the event that the adversary’s output is equal to b, then we define Xadv⇤ [A, S] := Pr[W ]
1/2
to be A’s “bitguessing X advantage.” Using exactly the same calculation as in the proof of Theorem 2.10, we have Xadv[A, S] = 2 · Xadv⇤ [A, S].
2.4
(2.13)
Mathematical details
Up until now, we have used the terms efficient and negligible rather loosely, without a formal mathematical definition: • we required that a computational cipher have efficient encryption and decryption algorithms; • for a semantically secure cipher, we required that any efficient adversary have a negligible advantage in Attack Game 2.1. The goal of this section is to provide precise mathematical definitions for these terms. While these definitions lead to a satisfying theoretical framework for the study of cryptography as a mathematical discipline, we should warn the reader: • the definitions are rather complicated, requiring an unfortunate amount of notation; and • the definitions model our intuitive understanding of these terms only very crudely. 27
We stress that the reader may safely skip this section without su↵ering a significant loss in understanding. Before marching headlong into the formal definitions, let us remind the reader of what we are trying to capture in these definitions. • First, when we speak of an efficient encryption or decryption algorithm, we usually mean one that runs very quickly, encrypting data at a rate of, say, 10–100 computer cycles per byte of data. • Second, when we speak of an efficient adversary, we usually mean an algorithm that runs in some large, but still feasible amount of time (and other resources). Typically, one assumes that an adversary that is trying to break a cryptosystem is willing to expend many more resources than a user of the cryptosystem. Thus, 10,000 computers running in parallel for 10 years may be viewed as an upper limit on what is feasibly computable by a determined, patient, and financially wello↵ adversary. However, in some settings, like the Internet roulette example in Section 2.3.4, the adversary may have a much more limited amount of time to perform its computations before they become irrelevant. • Third, when we speak of an adversary’s advantage as being negligible, we mean that it is so small that it may as well be regarded as being equal to zero for all practical purposes. As we saw in the Internet roulette example, if no efficient adversary has an advantage better than 2 100 in Attack Game 2.1, then no player can in practice improve his odds at winning Internet roulette by more than 2 100 relative to physical roulette. Even though our intuitive understanding of the term efficient depends on the context, our formal definition will not make any such distinction. Indeed, we shall adopt the computational complexity theorist’s habit of equating the notion of an efficient algorithm with that of a (probabilistic) polynomialtime algorithm. For better and for worse, this gives us a formal framework that is independent of the specific details of any particular model of computation.
2.4.1
Negligible, superpoly, and polybounded functions
We begin by defining the notions of negligible, superpoly, and polybounded functions. Intuitively, a negligible function f : Z 0 ! R is one that not only tends to zero as n ! 1, but does so faster than the inverse of any polynomial. Definition 2.5. A function f : Z n0 2 Z 1 such that for all integers n
! R is called negligible if for all c 2 R>0 there exists n0 , we have f (n) < 1/nc .
1
An alternative characterization of a negligible function, which is perhaps easier to work with, is the following: Theorem 2.11. A function f : Z
1
! R is negligible if and only if for all c > 0, we have lim f (n)nc = 0.
n!1
Proof. Exercise. 2 Example 2.10. Some examples of negligible functions: 2
n
, 2
p
n
, n
28
log n
.
Some examples of nonnegligible functions: 1 1 , . 1000n4 + n2 log n n100
2
Once we have the term “negligible” formally defined, defining “superpoly” is easy: Definition 2.6. A function f : Z
1
! R is called superpoly if 1/f is negligible.
Essentially, a polybounded function f : Z some polynomial. Formally:
1
! R is one that is bounded (in absolute value) by
Definition 2.7. A function f : Z 1 ! R is called polybounded, if there exists c, d 2 R>0 such that for all integers n 0, we have f (n) nc + d. Note that if f is a polybounded function, then 1/f is definitely not a negligible function. However, as the following example illustrates, one must take care not to draw erroneous inferences. Example 2.11. Define f : Z 1 ! R so that f (n) = 1/n for all even integers n and f (n) = 2 n for all odd integers n. Then f is not negligible, and 1/f is neither polybounded nor superpoly. 2
2.4.2
Computational ciphers: the formalities
Now the formalities. We begin by admitting a lie: when we said a computational cipher E = (E, D) is defined over (K, M, C), where K is the key space, M is the message space, and C is the ciphertext space, and with each of these spaces being finite sets, we were not telling the whole truth. In the mathematical model (though not always in realworld systems), we associate with E families of key, message, and ciphertext spaces, indexed by • a security parameter, which is a positive integer, and is denoted by , and • a system parameter, which is a bit string, and is denoted by ⇤. Thus, instead of just finite sets K, M, and C, we have families of finite sets {K
,⇤ } ,⇤ ,
{M
,⇤ } ,⇤ ,
and
{C
,⇤ } ,⇤ ,
which for the purposes of this definition, we view as sets of bit strings (which may represent mathematical objects by way of some canonical encoding functions). The idea is that when the cipher E is deployed, the security parameter is fixed to some value. Generally speaking, larger values of imply higher levels of security (i.e., resistance against adversaries with more computational resources), but also larger key sizes, as well as slower encryption and decryption speeds. Thus, the security parameter is like a “dial” we can turn, setting a tradeo↵ between security and efficiency. Once is chosen, a system parameter ⇤ is generated using an algorithm specific to the cipher. The idea is that the system parameter ⇤ (together with ) gives a detailed description of a fixed instance of the cipher, with (K, M, C) = (K ,⇤ , M ,⇤ , C ,⇤ ). This one, fixed instance may be deployed in a larger system and used by many parties — the values of and ⇤ are public and known to everyone (including the adversary). 29
Example 2.12. Consider the additive onetime pad discussed in Example 2.4. This cipher was described in terms of a modulus n. To deploy such a cipher, a suitable modulus n is generated, and is made public (possibly just “hardwired” into the software that implements the cipher). The modulus n is the system parameter for this cipher. Each specific value of the security parameter determines the length, in bits, of n. The value n itself is generated by some algorithm that may be probabilistic and whose output distribution may depend on the intended application. For example, we may want to insist that n is a prime in some applications. 2 Before going further, we define the notion of an efficient algorithm. For the purposes of this definition, we shall only consider algorithms A that take as input a security parameter , as well as other parameters whose total length is bounded by some fixed polynomial in . Basically, we want to say that the running time of A is bounded by a polynomial in , but things are complicated if A is probabilistic: Definition 2.8 (efficient algorithm). Let A be a an algorithm (possibly probabilistic) that takes as input a security parameter 2 Z 1 , as well as other parameters encoded as a bit string x 2 {0, 1}p( ) for some fixed polynomial p. We call A an efficient algorithm if there exist a polybounded function t and a negligible function ✏ such that for all 2 Z 1 , and all x 2 {0, 1}p( ) , the probability that the running time of A on input ( , x) exceeds t( ) is at most ✏( ). We stress that the probability in the above definition is with respect to the coin tosses of A: this bound on the probability must hold for all possible inputs x.1 Here is a formal definition that captures the basic requirements of systems that are parameterized by a security and system parameter, and introduces some more terminology. In the following definition we use the notation Supp(P ( )) to refer to the support of the distribution P ( ), which is the set of all possible outputs of algorithm P on input . Definition 2.9. A system parameterization is an efficient probabilistic algorithm P that given a security parameter 2 Z 1 as input, outputs a bit string ⇤, called a system parameter, whose length is always bounded by a polynomial in . We also define the following terminology: • A collection S = {S ,⇤ } ,⇤ of finite sets of bits strings, where runs over Z 1 and ⇤ runs over Supp(P ( )), is called a family of spaces with system parameterization P , provided the lengths of all the strings in each of the sets S ,⇤ are bounded by some polynomial p in . • We say that S is efficiently recognizable if there is an efficient deterministic algorithm that on input 2 Z 1 , ⇤ 2 Supp(P ( )), and s 2 {0, 1}p( ) , determines if s 2 S ,⇤ . • We say that S is efficiently sampleable if there is an efficient probabilistic algorithm that on input 2 Z 1 and ⇤ 2 Supp(P ( )), outputs an element uniformly distributed over S ,⇤ . 1
By not insisting that a probabilistic algorithm halts in a specified time bound with probability 1, we give ourselves a little “wiggle room,” which allows us to easily do certain types of random sampling procedure that have no a priori running time bound, but are very unlikely to run for too long (e.g., think of flipping a coin until it comes up “heads”). An alternative approach would be to bound the expected running time, but this turns out to be somewhat problematic for technical reasons. Note that this definition of an efficient algorithm does not require that the algorithm halt with probability 1 on all inputs. An algorithm that with probability 2 entered an infinite loop would satisfy the definition, even though it does not halt with probability 1. These issues are rather orthogonal. In general, we shall only consider algorithms that halt with probability 1 on all inputs: this can more naturally be seen as a requirement on the output distribution of the algorithm, rather than on its running time.
30
• We say that S has an e↵ective length function if there is an efficient deterministic algorithm that on input 2 Z 1 , ⇤ 2 Supp(P ( )), and s 2 S ,⇤ , outputs a nonnegative integer, called the length of s. We can now state the complete, formal definition of a computational cipher: Definition 2.10 (computational cipher). A computational cipher consists of a pair of algorithms E and D, along with three families of spaces with system parameterization P : K = {K
,⇤ } ,⇤ ,
M = {M
,⇤ } ,⇤ ,
and
C = {C
,⇤ } ,⇤ ,
such that 1. K, M, and C are efficiently recognizable. 2. K is efficiently sampleable. 3. M has an e↵ective length function. 4. Algorithm E is an efficient probabilistic algorithm that on input , ⇤, k, m, where ⇤ 2 Supp(P ( )), k 2 K ,⇤ , and m 2 M ,⇤ , always outputs an element of C ,⇤ .
2Z
1,
5. Algorithm D is an efficient deterministic algorithm that on input , ⇤, k, c, where 2 Z 1 , ⇤ 2 Supp(P ( )), k 2 K ,⇤ , and c 2 C ,⇤ , outputs either an element of M ,⇤ , or a special symbol reject 2 / M ,⇤ . 6. For all , ⇤, k, m, c, where 2 Z 1 , ⇤ 2 Supp(P ( )), k 2 K Supp(E( , ⇤; k, m)), we have D( , ⇤; k, c) = m.
,⇤ ,
m 2 M
,⇤ ,
and c 2
Note that in the above definition, the encryption and decryption algorithms take and ⇤ as auxiliary inputs. So as to be somewhat consistent with the notation already introduced in Section 2.3.1, we write this as E( , ⇤; · · · ) and D( , ⇤; · · · ). Example 2.13. Consider the additive onetime pad (see Example 2.12). In our formal framework, the security parameter determines the bit length L( ) of the modulus n, which is the system parameter. The system parameter generation algorithm takes as input and generates a modulus n of length L( ). The function L(·) should be polynomially bounded. With this assumption, it is clear that the system parameter generation algorithm satisfies its requirements. The requirements on the key, message, and ciphertext spaces are also satisfied: 1. Elements of these spaces have polynomially bounded lengths: this again follows from our assumption that L(·) is polynomially bounded. 2. The key space is efficiently sampleable: just choose k
R
{0, . . . , n
1}.
3. The key, message, and ciphertext spaces are efficiently recognizable: just test if a bit string s is the binary encoding of an integer between 0 and n 1. 4. The message space also has an e↵ective length function: just output (say) 0.
2
We note that some ciphers (for example the onetime pad) may not need a system parameter. In this case, we can just pretend that the system parameter is, say, the empty string. We also note that some ciphers do not really have a security parameter either; indeed, many industrystandard ciphers simply come readymade with a fixed key size, with no security parameter that can be tuned. This is simply mismatch between theory and practice — that is just the way it is. 31
That completes our formal mathematical description of a computational cipher, in all its glorious detail.2 The reader should hopefully appreciate that while these formalities may allow us to make mathematically precise and meaningful statements, they are not very enlightening, and mostly serve to obscure what is really going on. Therefore, in the main body of the text, we will continue to discuss ciphers using the simplified terminology and notation of Section 2.3.1, with the understanding that all statements made have a proper and natural interpretation in the formal framework discussed in this section. This will be a pattern that is repeated in the sequel: we shall mainly discuss various types of cryptographic schemes using a simplified terminology, without mention of security parameters and system parameters — these mathematical details will be discussed in a separate section, but will generally follow the same general pattern established here.
2.4.3
Efficient adversaries and attack games
In defining the notion of semantic security, we have to define what we mean by an efficient adversary. Since this concept will be used extensively throughout the text, we present a more general framework here. For any type of cryptographic scheme, security will be defined using an attack game, played between an adversary A and a challenger: A follows an arbitrary protocol, while the challenger follows some simple, fixed protocol determined by the cryptographic scheme and the notion of security under discussion. Furthermore, both adversary and challenger take as input a common security parameter , and the challenger starts the game by computing a corresponding system parameter ⇤, and sending this to the adversary. To model these types of interactions, we introduce the notion of an interactive machine. Before such a machine M starts, it always gets the security parameter written in a special bu↵er, and the rest of its internal state is initialized to some default value. Machine M has two other special bu↵ers: an incoming message bu↵er and an outgoing message bu↵er. Machine M may be invoked many times: each invocation starts when M ’s external environment writes a string to M ’s incoming message bu↵er; M reads the message, performs some computation, updates its internal state, and writes a string on its outgoing message bu↵er, ending the invocation, and the outgoing message is passed to the environment. Thus, M interacts with its environment via a simple message passing system. We assume that M may indicate that it has halted by including some signal in its last outgoing message, and M will essentially ignore any further attempts to invoke it. We shall assume messages to and from the machine M are restricted to be of constant length. This is not a real restriction: we can always simulate the transmission of one long message by sending many shorter ones. However, making a restriction of this type simplifies some of the technicalities. We assume this restriction from now on, for adversaries as well as for any other type of interactive machine. For any given environment, we can measure the total running time of M by counting the number of steps it performs across all invocations until it signals that it has halted. This running time depends not only on M and its random choices, but also on the environment in which M runs.3 2
Note that the definition of a Shannon cipher in Section 2.2.1 remains unchanged. The claim made at the end of Section 2.3.1 that any deterministic computational cipher is also a Shannon cipher needs to be properly interpreted: for each and ⇤, we get a Shannon cipher defined over (K ,⇤ , M ,⇤ , C ,⇤ ). 3 Analogous to the discussion in footnote 1 on page 30, our definition of an efficient interactive machine will not require that it halts with probability 1 for all environments. This is an orthogonal issue, but it will be an implicit
32
Definition 2.11 (efficient interactive machine). We say that M is an efficient interactive machine if there exist a polybounded function t and a negligible function ✏, such that for all environments (not even computationally bounded ones), the probability that the total running time of M exceeds t( ) is at most ✏( ). We naturally model an adversary as an interactive machine. An efficient adversary is simply an efficient interactive machine. We can connect two interactive machines together, say M 0 and M , to create a new interactive machine M 00 = hM 0 , M i. Messages from the environment to M 00 always get routed to M 0 . The machine M 0 may send a message to the environment, or to M ; in the latter case, the output message sent by M gets sent to M 0 . We assume that if M halts, then M 0 does not send it any more messages. See Fig. ??. Thus, when M 00 is invoked, its incoming message is routed to M 0 , and then M 0 and M may interact some number of times, and then the invocation of M 00 ends when M 0 sends a message to the environment. We call M 0 the “open” machine (which interacts with the outside world), and M the “closed” machine (which interacts only with M 0 ). Naturally, we can model the interaction of a challenger and an adversary by connecting two such machines together as above: the challenger becomes the open machine, and the adversary becomes the closed machine. In our security reductions, we typically show how to use an adversary A that breaks some system to build an adversary B that breaks some other system. The essential property that we want is that if A is efficient, then so is B. However, our reductions are almost always of a very special form, where B is a wrapper around A, consisting of some simple and efficient “interface layer” between B’s challenger and a single running instance of A. Ideally, we want the computational complexity of the interface layer to not depend on the computational complexity of A; however, some dependence is unavoidable: the more queries A makes to its challenger, the more work must be performed by the interface layer, but this work should just depend on the number of such queries and not on the running time of A. To formalize this, we build B as a composed machine hM 0 , M i, where M 0 represents the interface layer (the “open” machine), and M represents the instance of A (the “closed” machine). This leads us to the following definition. Definition 2.12 (elementary wrapper). An interactive machine M 0 is called an efficient interface if there exists a polybounded function t and a negligible function ✏, such that for all M (not necessarily computationally bounded), when we execute the composed machine hM 0 , M i in an arbitrary environment (again, not necessarily computationally bounded), the following property holds: at every point in the execution of hM 0 , M i, if I is the number of interactions between M 0 and M up to at that point, and T is the total running time of M 0 up to that point, then the probability that T > t( + I) is at most ✏( ). If M 0 is an efficient interface, and M is any machine, then we say hM 0 , M i is an elementary wrapper around M . requirement of any machines we consider.
33
Thus, we will say adversary B is an elementary wrapper around adversary A when it can be structured as above, as an efficient interface interacting with A. Our definitions were designed to work well together. The salient properties are: • If B is an elementary wrapper around A, and A is efficient, then B is efficient. • If C is an elementary wrapper around B and B is an elementary wrapper around A, then C is an elementary wrapper around A. Also note that in our attack games, the challenger is typically satisfies our definition of an efficient interface. For such a challenger and any efficient adversary A, we can view their entire interaction as a that of a single, efficient machine. Query bounded adversaries. In the attack games we have seen so far, the adversary makes just a fixed number of queries. Later in the text, we will see attack games in which the adversary A is allowed to make many queries — even though there is no a priori bound on the number of queries it is allowed to make, if A is efficient, the number of queries will be bounded by some polybounded value Q (at least with all but negligible probability). In proving security for such attack games, in designing an elementary wrapper B from A, it will usually be convenient to tell B in advance an upper bound Q on how many queries A will ultimately make. To fit this into our formal framework, we can set things up so that A starts out by sending a sequence of Q special messages to “signal” this query bound to B. If we do this, then not only can B use the value Q in its logic, it is also allowed to run in time that depends on Q, without violating the time constraints in Definition 2.12. This is convenient, as then B is allowed to initialize data structures whose size may depend on Q. Of course, all of this is just a legalistic “hack” to work around technical constraints that would otherwise be too restrictive, and should not be taken too seriously. We will never make this “signaling” explicit in any of our presentations.
2.4.4
Semantic security: the formalities
In defining any type of security, we will define the adversary’s advantage in the attack game as a function Adv( ). This will be defined in terms of probabilities of certain events in the attack game: for each value of we get a di↵erent probability space, determined by the random choices of the challenger, and the random choices made the adversary. Security will mean that for every efficient adversary, the function Adv(·) is negligible. Turning now to the specific situation of semantic security of a cipher, in Attack Game 2.1, we defined the value SSadv[A, E]. This value is actually a function of the security parameter . The proper interpretation of Definition 2.3 is that E is secure if for all efficient adversaries A (modeled as an interactive machine, as described above), the function SSadv[A, E]( ) in the security parameter is negligible (as defined in Definition 2.5). Recall that both challenger and adversary receive as a common input. Control begins with the challenger, who sends the system parameter to the adversary. The adversary then sends its query to the challenger, which consists of two plaintexts, who responds with a ciphertext. Finally, the adversary outputs a bit (technically, in our formal machine model, this “output” is a message sent to the challenger, and then the challenger halts). The value of SSadv[A, E]( ) is determined by the random choices of the challenger (including the choice of system parameter) and the random choices of the adversary. See Fig. 2.6 for a complete picture of Attack Game 2.1. 34
A
Challenger (Experiment b)
⇤ k c
R
R
R
⇤
P( )
K , E( , ⇤; k, mb )
m0 , m1 2 M
,
c ˆb 2 {0, 1}
Figure 2.6: The fully detailed version of Attack Game 2.1 Also, in Attack Game 2.1, the requirement that the two messages presented by the adversary have the same length means that the length function provided in part 3 of Definition 2.10 evaluates to the same value on the two messages. It is perhaps useful to see what it means for a cipher E to be insecure according to this formal definition. This means that there exists an adversary A such that SSadv[A, E] is a nonnegligible function in the security parameter. This means that SSadv[A, E]( ) 1/ c for some c > 0 and for infinitely many values of the security parameter . So this does not mean that A can “break” E for all values of the security parameter, but only infinitely many values of the security parameter. In the main body of the text, we shall mainly ignore security parameters, system parameters, and the like, but it will always be understood that all of our “shorthand” has a precise mathematical interpretation. In particular, we will often refer to certain values v as be negligible (resp., polybounded), which really means that v is a negligible (resp., polybounded) function of the security parameter.
2.5
A fun application: anonymous routing
Our friend Alice wants to send a message m to Bob, but she does not want Bob or anyone else to know that the message m is from Alice. For example, Bob might be running a public discussion forum and Alice wants to post a comment anonymously on the forum. Posting anonymously lets Alice discuss health issues or other matters without identifying herself. In this section we will assume Alice only wants to post a single message to the forum. One option is for Alice to choose a proxy, Carol, send m to Carol, and ask Carol to forward the message to Bob. This clearly does not provide anonymity for Alice since anyone watching the network will see that m was sent from Alice to Carol and then from Carol to Bob. By tracing the 35
path of m through the network anyone can see that the post came from Alice. A better approach is for Alice to establish a shared key k with Carol and send c := E(k, m) to Carol, where E = (E, D) is a semantically secure cipher. Carol decrypts c and forwards m to Bob. Now, someone watching the network will see one message sent from Alice to Carol and a di↵erent message sent from Carol to Bob. Nevertheless, this method still does not ensure anonymity for Alice: if on a particular day the only message that Carol receives is the one from Alice and the only message she sends goes to Bob, then an observer can link the two and still learn that the posted message came from Alice. We solve this problem by having Carol provide a mixing service, that is, a service that mixes incoming messages from many di↵erent parties A1 , . . . , An . For i = 1, . . . , n, Carol establishes a secret key ki with party Ai and each party Ai sends to Carol an encrypted message ci := E ki , hdestinationi , mi i . Carol collects all n incoming ciphertexts, decrypts each of them with the correct key, and forwards the resulting plaintexts in some random order to their destinations. Now an observer examining Carol’s traffic sees n messages going in and n messages going out, but cannot tell which message was sent where. Alice’s message is one of the n messages sent out by Carol, but the observer cannot tell which one. We say that Alice’s anonymity set is of size n. The remaining problem is that Carol can still tell that Alice is the one who posted a specific message on the discussion forum. To eliminate this final risk Alice uses multiple mixing services, say, Carol and David. She establishes a secret key kc with Carol and a secret key kd with David. To send her message to Bob she constructs the following nested ciphertext c2 : c2 := E kc , E(kd , m) .
(2.14)
For completeness Alice may want to embed routing information inside the ciphertext so that c2 is actually constructed as: c2 := E kc , hDavid, c1 i
where
c1 := E kd , hBob, mi .
Next, Alice sends c2 to Carol. Carol decrypts c2 and obtains the plaintext hDavid, c1 i which tells her to send c1 to David. David decrypts c1 and obtains the plaintext hBob, mi which tells him to send m to Bob. This process of decrypting a nested ciphertext, illustrated in Fig. 2.7, is similar to peeling an onion one layer at a time. For this reason this routing procedure is often called onion routing. Now even if Carol observes all network traffic she cannot tell with certainty who posted a particular message on Bob’s forum. The same holds for David. However, if Carol and David collude they can figure it out. For this reason Alice may want to route her message through more than two mixes. As long as one of the mixes does not collude with the others, Alice’s anonymity will be preserved. One small complication is that when Alice establishes her shared secret key kd with David, she must do so without revealing her identity to David. Otherwise, David will know that c1 came from Alice, which we do not want. This is not difficult to do, and we will see how later in the book (Section 20.14). Security of nested encryption. To preserve Alice’s anonymity it is necessary that Carol, who knows kc , learn no information about m from the nested ciphertext c2 in (2.14). Otherwise, Carol could potentially use the information she learns about m from c2 to link Alice to her post on Bob’s discussion forum. For example, suppose Carol could learn the first few characters of m from c2 and 36
Alice&
mix&
c2&
Carol&
c1&
mix&
m&
David&
Bob&
Figure 2.7: An example onion routing using two mixes later find that there is only one post on Bob’s forum starting with those characters. Carol could then link the entire post to Alice because she knows that c2 came from Alice. The same holds for David: it had better be the case that David, who knows kd , can learn no information about m from the nested ciphertext c2 in (2.14). Let us argue that if E is semantically secure then no efficient adversary can learn any information about m given c2 and one of kc or kd . More generally, for a cipher E = (E, D) defined over (K, M, C) let us define the nway nested cipher En = (En , Dn ) as En (k0 , . . . , kn
1 ),
m = E kn
1,
E(kn
2,
· · · E(k0 , m) · · · ) .
Decryption applies the keys in the reverse order: Dn (k0 , . . . , kn
1 ),
c = D k0 , D(k1 , · · · D(kn
1 , c) · · · )
.
Our goal is to show that if E is semantically secure then En is semantically secure even if the adversary is given all but one of the keys k0 , . . . , kn 1 . To make this precise, we define two experiments, Experiment 0 and Experiment 1, where for b = 0, 1, Experiment b is: • The adversary gives the challenger (m0 , m1 , d) where m0 , m1 2 M are equal length messages and 0 d < n. • The challenger chooses n keys k0 , . . . , kn 1 R K and computes c R En (k0 , . . . , kn 1 ), mb . It sends c to the adversary along with all keys k0 , . . . , kn 1 , but excluding the key kd . • The adversary outputs a bit ˆb 2 {0, 1}. This game captures the fact that the adversary sees all keys k0 , . . . , kn 1 except for kd and tries to break semantic security. We define the adversary’s advantage, NE(n) adv[A, E], as in the definition of semantic security: NE(n) adv[A, E] = Pr[W0 ]
Pr[W1 ]
where Wb is the event that A outputs 1 in Experiment b, for b = 0, 1. We say that E is semantically secure for nway nesting if NE(n) adv[A, E] is negligible. Theorem 2.12. For every constant n > 0, if E = (E, D) is semantically secure then E is semantically secure for nway nesting. In particular, for every nway nested adversary A attacking En , there exists a semantic security adversary B attacking E, where B is an elementary wrapper around A, such that NE(n) adv[A, E] = SSadv[B, E] .
The proof of this theorem is a good exercise in security reductions. We leave it for Exercise 2.15. 37
2.6
Notes
The one time pad is due to Gilbert Vernam in 1917, although there is evidence that it was discovered earlier [10]. Citations to the literature to be added.
2.7
Exercises
2.1 (multiplicative onetime pad). We may also define a “multiplication mod p” variation of the onetime pad. This is a cipher E = (E, D), defined over (K, M, C), where K := M := C := {1, . . . , p 1}, where p is a prime. Encryption and decryption are defined as follows: E(k, m) := k · m mod p
D(k, c) := k
1
· c mod p.
Here, k 1 denotes the multiplicative inverse of k modulo p. Verify the correctness property for this cipher and prove that it is perfectly secure. 2.2 (A good substitution cipher). Consider a variant of the substitution cipher E = (E, D) defined in Example 2.3 where every symbol of the message is encrypted using an independent permutation. That is, let M = C = ⌃L for some a finite alphabet of symbols ⌃ and some L. Let the key space be K = S L where S is the set of all permutations on ⌃. The encryption algorithm E(k, m) is defined as E(k, m) :=
k[0](m[0]), k[1](m[1]), . . . , k[L
1](m[L
1])
Show that E is perfectly secure. 2.3 (Chain encryption). Let E = (E, D) be a perfectly secure cipher defined over (K, M, C) where K = M. Let E 0 = (E 0 , D0 ) be a cipher where encryption is defined as E 0 ((k1 , k2 ), m) := E(k1 , k2 ), E(k2 , m) . Show that E 0 is perfectly secure. 2.4 (A broken onetime pad). Consider a variant of the one time pad with message space {0, 1}L where the key space K is restricted to all Lbit strings with an even number of 1’s. Give an efficient adversary whose semantic security advantage is 1. 2.5 (A stronger impossibility result). This exercise generalizes Shannon’s theorem (Theorem 2.5). Let E be a cipher defined over (K, M, C). Suppose that SSadv[A, E] ✏ for all adversaries A, even including computationally unbounded ones. Show that K (1 ✏)M. 2.6 (A matching bound). This exercise develops a converse of sorts for the previous exercise. For j = 0, . . . , L 1, let ✏ = 1/2j . Consider the Lbit onetime pad variant E defined over (K, M, C) where M = C = {0, 1}L . The key space K is restricted to all Lbit strings whose first L j bits are not all zero, so that K = (1 ✏)M. Show that: (a) there is an efficient adversary A such that SSadv[A, E] = ✏/(1
✏);
(b) for all adversaries A, even including computationally unbounded ones, SSadv[A, E] ✏/(1 ✏). Note: Since the advantage of A in part (a) is nonzero, the cipher E cannot be perfectly secure. 38
2.7 (Deterministic ciphers). In this exercise, you are asked to prove in detail the claims made in Example 2.9. Namely, show that if E is a deterministic cipher that is perfectly secure, then SSadv[A, E] = 0 for every adversary A (bearing in mind that A may be probabilistic); also show that if E is the variable length onetime pad, then SSadv[A, E] = 0 for all adversaries A. 2.8 (Roulette). In Section 2.3.4, we argued that if value r is encrypted using a semantically secure cipher, then a player’s odds of winning at Internet roulette are very close to those of real roulette. However, our “roulette” game was quite simple. Suppose that we have a more involved game, where di↵erent outcomes may result in di↵erent winnings. The rules are not so important, but assume that the rules are easy to evaluate (given a bet and the number r) and that every bet results in a payout of 0, 1, . . . , n dollars, where n is polybounded. Let µ be the expected winnings in an optimal strategy for a real version of this game (with no encryption). Let µ0 be the expected winnings of some (efficient) player in an Internet version of this game (with encryption). Show that µ µ0 + ✏, where ✏ is negligible, assuming the cipher is semantically secure. Hint: You may want to use the fact that if XPis a random variable taking values in the set {0, 1, . . . , n}, the expected value of X is equal to ni=1 Pr[X i]. 2.9. Prove Fact 2.6, using the formal definitions in Section 2.4.
2.10 (Exercising the definition of semantic security). Let E = (E, D) be a semantically secure cipher defined over (K, M, C), where M = C = {0, 1}L . Which of the following encryption algorithms yields a semantically secure scheme? Either give an attack or provide a security proof via an explicit reduction. (a) E 0 (k, m) = 0 k E(k, m) (b) E 0 (k, m) = E(k, m) k parity(m) (c) E 0 (k, m) = reverse(E(k, m)) (d) E 0 (k, m) = E(k, reverse(m)) Here, for a bit string s, parity(s) is 1 if the number of 1’s in s is odd, and 0 otherwise; also, reverse(s) is the string obtained by reversing the order of the bits in s, e.g., reverse(1011) = 1101. 2.11 (Key recovery attacks). Let E = (E, D) be a cipher defined over (K, M, C). A key recovery attack is modeled by the following game between a challenger and an adversary A: the challenger chooses a random key k in K, a random message m in M, computes c R E(k, m), and sends (m, c) ˆ c) = m and define to A. In response A outputs a guess kˆ in K. We say that A wins the game if D(k, KRadv[A, E] to be the probability that A wins the game. As usual, we say that E is secure against key recovery attacks if for all efficient adversaries A the advantage KRadv[A, E] is negligible. (a) Show that the onetime pad is not secure against key recovery attacks. (b) Show that if E is semantically secure and ✏ = K/M is negligible, then E is secure against key recovery attacks. In particular, show that for every efficient keyrecovery adversary A there is an efficient semantic security adversary B, where B is an elementary wrapper around A, such that KRadv[A, E] SSadv[B, E] + ✏ 39
Hint: Your semantic security adversary B will output 1 with probability KRadv[A, E] in the semantic security Experiment 0 and output 1 with probability at most ✏ in Experiment 1. Deduce from this a lower bound on SSadv[B, E] in terms of ✏ and KRadv[A, E] from which the result follows. (c) Deduce from part (b) that if E is semantically secure and M is superpoly then K cannot be polybounded. Note: K can be polybounded when M is polybounded, as in the onetime pad. 2.12 (Security against message recovery). In Section 2.3.3 we developed the notion of security against message recovery. Construct a cipher that is secure against message recovery, but is not semantically secure. 2.13 (Advantage calculations in simple settings). Consider the following two experiments Experiment 0 and Experiment 1: • In Experiment 0 the challenger flips a fair coin (probability 1/2 for HEADS and 1/2 for TAILS) and sends the result to the adversary A. • In Experiment 1 the challenger always sends TAILS to the adversary. The adversary’s goal is to distinguish these two experiments: at the end of each experiment the adversary outputs a bit 0 or 1 for its guess for which experiment it is in. For b = 0, 1 let Wb be the event that in experiment b the adversary output 1. The adversary tries to maximize its distinguishing advantage, namely the quantity Pr[W0 ]
Pr[W1 ]
2 [0, 1] .
If the advantage is negligible for all efficient adversaries then we say that the two experiments are indistinguishable. (a) Calculate the advantage of each of the following adversaries: (i) A1 : Always output 1.
(ii) A2 : Ignore the result reported by the challenger, and randomly output 0 or 1 with even probability. (iii) A3 : Output 1 if HEADS was received from the challenger, else output 0. (iv) A4 : Output 0 if HEADS was received from the challenger, else output 1.
(v) A5 : If HEADS was received, output 1. If TAILS was received, randomly output 0 or 1 with even probability.
(b) What is the maximum advantage possible in distinguishing these two experiments? Explain why. 2.14 (Permutation cipher). Consider the following cipher (E, D) defined over (K, M, C) where C = M = {0, 1}` and K is the set of all `! permutations of the set {0, . . . , ` 1}. For a key k 2 K and message m 2 M define E(k, m) to be result of permuting the bits of m using the permutation k, namely E(k, m) = m[k(0)]...m[k(` 1)]. Show that this cipher is not semantically secure by showing an adversary that achieves advantage 1. 40
2.15 (Nested encryption). For a cipher E = (E, D) define the nested cipher E 0 = (E 0 , D0 ) as E 0 (k0 , k1 ), m = E k1 , E(k0 , m)
and
D0 (k0 , k1 ), c = D(k0 , D(k1 , c)) .
Our goal is to show that if E is semantically secure then E 0 is semantically secure even if the adversary is given one of the keys k0 or k1 . (a) Consider the following semantic security experiments, Experiments 0 and 1: in Experiment b, for b = 0, 1, the adversary generates two messages m0 and m1 and gets back k1 and E 0 (k0 , k1 ), mb ). The adversary outputs ˆb in {0, 1} and we define its advantage, NEadv[A, E] as in the usual the definition of semantic security. Show that for every nested encryption adversary A attacking E 0 , there exists a semantic security adversary B attacking E, where B is an elementary wrapper around A, such that NEadv[A, E] = SSadv[B, E] . Draw a diagram with A on the right, B in the middle, and B’s challenger on the left. Show the message flow between these three parties that takes place in your proof of security. (b) Repeat part (a), but now when the adversary gets back k0 (instead of k1 ) and E 0 (k0 , k1 ), mb ) in Experiments 0 and 1. Draw a diagram describing the message flow in your proof of security as you did in part (a). This problem comes up in the context of anonymous routing on the Internet as discussed in Section 2.5. 2.16 (Self referential encryption). Let us show that encrypting a key under itself can be dangerous. Let E be a semantically secure cipher defined over (K, M, C), where K ✓ M, and let k R K. A ciphertext c⇤ := E(k, k), namely encrypting k using k, is called a self referential encryption. ˜ D) ˜ derived from E such that E˜ is semantically secure, but becomes (a) Construct a cipher E˜ = (E, ˜ k). You have just shown that semantic security does insecure if the adversary is given E(k, not imply security when one encrypts one’s key. ˆ D) ˆ derived from E such that Eˆ is semantically and remains (b) Construct a cipher Eˆ = (E, ˆ k). To prove that Eˆ is semantically secure (provably) even if the adversary is given E(k, ˆ semantically secure, you should show the following: for every adversary A that attacks E, there exists and adversary B that attacks E such that (i) the running time B is about the ˆ SSadv[B, E]. same as that of A, and (ii) SSadv[A, E] 2.17 (Compression and encryption). Two standards committees propose to save bandwidth by combining compression (such as the LempelZiv algorithm used in the zip and gzip programs) with encryption. Both committees plan on using the variable length one time pad for encryption. • One committee proposes to compress messages before encrypting them. Explain why this is a bad idea. Hint: Recall that compression can significantly shrink the size of some messages while having little impact on the length of other messages. • The other committee proposes to compress ciphertexts after encryption. Explain why this is a bad idea. 41
Over the years many problems have surfaced when combining encryption and compression. The CRIME [92] and BREACH [88] attacks are good representative examples. 2.18 (Voting protocols). This exercise develops a simple voting protocol based on the additive onetime pad (Example 2.4). Suppose we have t voters and a counting center. Each voter is going to vote 0 or 1, and the counting center is going to tally the votes and broadcast the total sum S. However, they will use a protocol that guarantees that no party (voter or counting center) learns anything other than S (but we shall assume that each party faithfully follows the protocol). The protocol works as follows. Let n > t be an integer. The counting center generates an encryption of 0: c0 R {0, . . . , n 1}, and passes c0 to voter 1. Voter 1 adds his vote v1 to c0 , computing c0 + v1 mod n, and passes c1 to voter 2. This continues, with each voter i adding vi to ci 1 , c1 computing ci ci 1 + vi mod n, and passing ci to voter i + 1, except that voter t passes ct to the counting center. The counting center computes the total sum as S ct c0 mod n, and broadcasts S to all the voters. (a) Show that the protocol correctly computes the total sum. (b) Show that the protocol is perfectly secure in the following sense. For voter i = 1, . . . , t, define View i := (S, ci 1 ), which represents the “view” of voter i. We also define View 0 := (c0 , ct ), which represents the “view” of the counting center. Show that for each i = 0, . . . , t and S = 0, . . . , t, the following holds: as the choice P of votes v1 , . . . , vt varies, subject to the restrictions that each vj 2 {0, 1} and tj=1 vj = S, the distribution of View i remains the same.
(c) Show that if two voters i, j collude, they can determine the vote of a third voter k. You are free to choose the indices i, j, k. 2.19 (Twoway split keys). Let E = (E, D) be a semantically secure cipher defined over (K, M, C) where K = {0, 1}d . Suppose we wish to split the ability to decrypt ciphertexts across two parties, Alice and Bob, so that both parties are needed to decrypt ciphertexts. For a random key k in K choose a random r in K and define ka := r and kb := k r. Now if Alice and Bob get together they can decrypt a ciphertext c by first reconstructing the key k as k = ka kb and then computing D(k, c). Our goal is to show that neither Alice nor Bob can decrypt ciphertexts on their own. (a) Formulate a security notion that captures the advantage that an adversary has in breaking semantic security given Bob’s key kb . Denote this 2way key splitting advantage by 2KSadv[A, E]. (b) Show that for every 2way key splitting adversary A there is a semantic security adversary B such that 2KSadv[A, E] = SSadv[B, E]. 2.20 (Simple secret sharing). Let E = (E, D) be a semantically secure cipher with key space K = {0, 1}L . A bank wishes to split a decryption key k 2 {0, 1}L into three shares p0 , p1 , and p2 so that two of the three shares are needed for decryption. Each share can be given to a di↵erent bank executive, and two of the three must contribute their shares for decryption to proceed. This way, decryption can proceed even if one of the executives is out sick, but at least two executives are needed for decryption. 42
(a) To do so the bank generates two random pairs (k0 , k00 ) and (k1 , k10 ) so that k0 k00 = k1 k10 = k. How should the bank assign shares so that any two shares enable decryption using k, but no single share can decrypt? Hint: The first executive will be given the share p0 := (k0 , k1 ). (b) Generalize the scheme from part (a) so that 3outof5 shares are needed for decryption. Reconstituting the key only uses XOR of key shares. Two shares should reveal nothing about the key k. (c) More generally, we can design a toutofw system this way for any t < w. How does the size of each share scale with t? We will see a much better way to do this in Section 11.6. 2.21 (Simple threshold decryption). Let E = (E, D) be a semantically secure cipher with key space K. In this exercise we design a system that lets a bank split a key k into three shares p0 , p1 , and p2 so that two of the three shares are needed for decryption, as in Exercise 2.20. However, decryption is done without ever reconstituting the complete key at a single location. We use nested encryption from Exercise 2.15. Choose a random key k := (k0 , k1 , k2 , k3 ) in K4 and encrypt a message m as: ✓ ◆ R c E k1 , E(k0 , m) , E k4 , E(k3 , m) . (a) Construct the shares p0 , p1 , p2 so that any two shares enable decryption, but no single share can decrypt. Hint: the first share is p0 := (k0 , k3 ). Discussion: Suppose the entities holding shares p0 and p2 are available to decrypt. To decrypt a ciphertext c, first send c to the entity holding p2 to partially decrypt c. Then forward the result to the entity holding p0 to complete the decryption. This way, decryption is done without reconstituting the complete key k at a single location. (b) Generalize the scheme from part (a) so that 3outof5 shares are needed for decryption. Explain how decryption can be done without reconstituting the key in a single location. An encryption scheme where the key can be split into shares so that toutofw shares are needed for decryption, and decryption does not reconstitute the key at a single location, is said to provide threshold decryption. We will see a much better way to do this in Section 11.6. 2.22 (Bias correction). Consider again the bitguessing version of the semantic security attack game (i.e., Attack Game 2.4). Suppose an efficient adversary A wins the game (i.e., guesses the hidden bit b) with probability 1/2 + ✏, where ✏ is nonnegligible. Note that ✏ could be positive or negative (the definition of negligible works on absolute values). Our goal is to show that there is another efficient adversary B that wins the game with probability 1/2+✏0 , where ✏0 is nonnegligible and positive. (a) Consider the following adversary B that uses A as a subroutine in Attack Game 2.4 in the following twostage attack. In the first stage, B plays challenger to A, but B generates its own hidden bit b0 , its own key k0 , and eventually A outputs its guessbit ˆb0 . Note that in this stage, B’s challenger in Attack Game 2.4 is not involved at all. In the second stage, B restarts A, and lets A interact with the “real” challenger in Attack Game 2.4, and eventually 43
A outputs a guessbit ˆb. When this happens, B outputs ˆb ˆb0 b0 . Note that this run of A is completely independent of the first — the coins of A and also the system parameters are generated independently in these two runs. Show that B wins Attack Game 2.4 with probability 1/2 + 2✏2 . (b) One might be tempted to argue as follows. Just construct an adversary B that runs A, and when A outputs ˆb, adversary B outputs ˆb 1. Now, we do not know if ✏ is positive or negative. If it is positive, then A satisfies are requirements. If it is negative, then B satisfies our requirements. Although we do not know which one of these two adversaries satisfies our requirements, we know that one of them definitely does, and so existence is proved. What is wrong with this argument? The explanation requires an understanding of the mathematical details regarding security parameters (see Section 2.4). (c) Can you come up with another efficient adversary B 0 that wins the bitguessing game with probability at least 1 + ✏/2? Your adversary B 0 will be less efficient than B.
44
Chapter 3
Stream ciphers In the previous chapter, we introduced the notions of perfectly secure encryption and semantically secure encryption. The problem with perfect security is that to achieve it, one must use very long keys. Semantic security was introduced as a weaker notion of security that would perhaps allow us to build secure ciphers that use reasonably short keys; however, we have not yet produced any such ciphers. This chapter studies one type of cipher that does this: the stream cipher.
3.1
Pseudorandom generators
Recall the onetime pad. Here, keys, messages, and ciphertexts are all Lbit strings. However, we would like to use a key that is much shorter. So the idea is to instead use a short, `bit “seed” s as the encryption key, where ` is much smaller than L, and to “stretch” this seed into a longer, Lbit string that is used to mask the message (and unmask the ciphertext). The string s is stretched using some efficient, deterministic algorithm G that maps `bit strings to Lbit strings. Thus, the key space for this modified onetime pad is {0, 1}` , while the message and ciphertext spaces are {0, 1}L . For s 2 {0, 1}` and m, c 2 {0, 1}L , encryption and decryption are defined as follows: E(s, m) := G(s)
m and
D(s, c) := G(s)
c.
This modified onetime pad is called a stream cipher, and the function G is called a pseudorandom generator. If ` < L, then by Shannon’s Theorem, this stream cipher cannot achieve perfect security; however, if G satisfies an appropriate security property, then this cipher is semantically secure. Suppose s is a random `bit string and r is a random Lbit string. Intuitively, if an adversary cannot e↵ectively tell the di↵erence between G(s) and r, then he should not be able to tell the di↵erence between this stream cipher and a onetime pad; moreover, since the latter cipher is semantically secure, so should be the former. To make this reasoning rigorous, we need to formalize the notion that an adversary cannot “e↵ectively tell the di↵erence between G(s) and r.” An algorithm that is used to distinguish a pseudorandom string G(s) from a truly random string r is called a statistical test. It takes a string as input, and outputs 0 or 1. Such a test is called e↵ective if the probability that it outputs 1 on a pseudorandom input is significantly di↵erent than the probability that it outputs 1 on a truly random input. Even a relatively small di↵erence in probabilities, say 1%, is considered significant; indeed, even with a 1% di↵erence, if we can obtain a few hundred independent samples, which are either all pseudorandom or all truly 45
random, then we will be able to infer with high confidence whether we are looking at pseudorandom strings or at truly random strings. However, a nonzero but negligible di↵erence in probabilities, say 2 100 , is not helpful. How might one go about designing an e↵ective statistical test? One basic approach is the following: given an Lbit string, calculate some statistic, and then see if this statistic di↵ers greatly from what one would expect if the string were truly random. For example, a very simple statistic that is easy to compute is the number k of 1’s appearing in the string. For a truly random string, we would expect k ⇡ L/2. If the PRG G had some bias towards either 0bits or 1bits, we could e↵ectively detect this with a statistical test that, say, outputs 1 if k 0.5L < 0.01L, and otherwise outputs 0. This statistical test would be quite e↵ective if the PRG G did indeed have some significant bias towards either 0 or 1. The test in the previous example can be strengthened by considering not just individual bits, but pairs of bits. One could break the Lbit string up into ⇡ L/2 bit pairs, and count the number k00 of pairs 00, the number k01 of pairs 01, the number k10 of pairs 10, and the number k11 of pairs 11. For a truly random string, one would expect each of these numbers to be ⇡ L/2 · 1/4 = L/8. Thus, a natural statistical test would be one that tests if the distance from L/8 of each of these numbers is less than some specified bound. Alternatively, one could sum up the squares of these distances, and test whether this sum is less than some specified bound — this is the classical squared test from statistics. Obviously, this idea generalizes from pairs of bits to tuples of any length. There are many other simple statistics one might check. However, simple tests such as these do not tend to exploit deeper mathematical properties of the algorithm G that a malicious adversary may be able to exploit in designing a statistical test specifically geared towards G. For example, there are PRG’s for which the simple tests in the previous two paragraphs are completely ine↵ective, but yet are completely predictable, given sufficiently many output bits; that is, given a prefix of G(s) of sufficient length, the adversary can compute all the remaining bits of G(s), or perhaps even compute the seed s itself. Our definition of security for a PRG formalizes the notion there should be no e↵ective (and efficiently computable) statistical test.
3.1.1
Definition of a pseudorandom generator
A pseudorandom generator, or PRG for short, is an efficient, deterministic algorithm G that, given as input a seed s, computes an output r. The seed s comes from a finite seed space S and the output r belongs to a finite output space R. Typically, S and R are sets of bit strings of some prescribed length (for example, in the discussion above, we had S = {0, 1}` and R = {0, 1}L ). We say that G is a PRG defined over (S, R). Our definition of security for a PRG captures the intuitive notion that if s is chosen at random from S and r is chosen at random from R, then no efficient adversary can e↵ectively tell the di↵erence between G(s) and r: the two are computationally indistinguishable. The definition is formulated as an attack game. Attack Game 3.1 (PRG). For a given PRG G, defined over (S, R), and for a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define: Experiment b:
• The challenger computes r 2 R as follows: 46
A
Challenger (Experiment 0)
s
R
r
S
G(s)
r ˆb 2 {0, 1}
A
Challenger (Experiment 1)
r
R
R
r ˆb 2 {0, 1}
Figure 3.1: Experiments 0 and 1 of Attack Game 3.1 – if b = 0: s
R
– if b = 1: r
R
S, r
G(s);
R.
and sends r to the adversary. • Given r, the adversary computes and outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to G as PRGadv[A, G] := Pr[W0 ] Pr[W1 ] . 2 The attack game is illustrated in Fig. 3.1. Definition 3.1 (secure PRG). A PRG G is secure if the value PRGadv[A, G] is negligible for all efficient adversaries A. As discussed in Section 2.3.5, Attack Game 3.1 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A. In this game, we measure A’s bitguessing advantage 47
PRGadv⇤ [A, G] as Pr[ˆb = b] here as well:
1/2. The general result of Section 2.3.5 (namely, (2.13)) applies PRGadv[A, G] = 2 · PRGadv⇤ [A, G].
(3.1)
We also note that a PRG can only be secure if the cardinality of the seed space is superpoly (see Exercise 3.5).
3.1.2
Mathematical details
Just as in Section 2.4, we give here more of the mathematical details pertaining to PRGs. Just like Section 2.4, this section may be safely skipped on first reading with very little loss in understanding. First, we state the precise definition of a PRG, using the terminology introduced in Definition 2.9. Definition 3.2 (pseudorandom generator). A pseudorandom generator consists of an algorithm G, along with two families of spaces with system parameterization P : S = {S
,⇤ } ,⇤
and
R = {R
,⇤ } ,⇤ ,
such that 1. S and R are efficiently recognizable and sampleable. 2. Algorithm G is an efficient deterministic algorithm that on input ⇤ 2 Supp(P ( )), and s 2 S ,⇤ , outputs an element of R ,⇤ .
, ⇤, s, where
2 Z
1,
Next, Definition 3.1 needs to be properly interpreted. First, in Attack Game 3.1, it is to be understood that for each value of the security parameter , we get a di↵erent probability space, determined by the random choices of the challenger and the random choices of the adversary. Second, the challenger generates a system parameter ⇤, and sends this to the adversary at the very start of the game. Third, the advantage PRGadv[A, G] is a function of the security parameter , and security means that this function is a negligible function.
3.2
Stream ciphers: encryption with a PRG
Let G be a PRG defined over ({0, 1}` , {0, 1}L ); that is, G stretches an `bit seed to an Lbit output. The stream cipher E = (E, D) constructed from G is defined over ({0, 1}` , {0, 1}L , {0, 1}L ); for s 2 {0, 1}` and m, c 2 {0, 1}L , encryption and decryption are defined as follows: if m = v, then E(s, m) := G(s)[0 . . v 1] m, and if c = v, then
D(s, c) := G(s)[0 . . v
1]
c.
As the reader may easily verify, this satisfies our definition of a cipher (in particular, the correctness property is satisfied). Note that for the purposes of analyzing the semantic security of E, the length associated with a message m in Attack Game 2.1 is the natural length m of m in bits. Also, note that if v is much smaller than L, then for many practical PRGs, it is possible to compute the first v bits of G(s) much faster than actually computing all the bits of G(s) and then truncating. The main result of this section is the following: 48
Theorem 3.1. If G is a secure PRG, then the stream cipher E constructed from G is a semantically secure cipher. In particular, for every SS adversary A that attacks E as in Attack Game 2.1, there exists a PRG adversary B that attacks G as in Attack Game 3.1, where B is an elementary wrapper around A, such that SSadv[A, E] = 2 · PRGadv[B, G]. (3.2)
Proof idea. The basic idea is to argue that we can replace the output of the PRG by a truly random string, without a↵ecting the adversary’s advantage by more than a negligible amount. However, after making this replacement, the adversary’s advantage is zero. 2 Proof. Let A be an efficient adversary attack E as in Attack Game 2.1. We want to show that SSadv[A, E] is negligible, assuming that G is a secure PRG. It is more convenient to work with the bitguessing version of the SS attack game. We prove: SSadv⇤ [A, E] = PRGadv[B, G]
(3.3)
for some efficient adversary B. Then (3.2) follows from Theorem 2.10. Moreover, by the assumption the G is a secure PRG, the quantity PRGadv[B, G] must negligible, and so the quantity SSadv[A, E] is negligible as well. So consider the adversary A’s attack of E in the bitguessing version of Attack Game 2.1. In this game, A presents the challenger with two messages m0 , m1 of the same length; the challenger then chooses a random key s and a random bit b, and encrypts mb under s, giving the resulting ciphertext c to A; finally, A outputs a bit ˆb. The adversary A wins the game if ˆb = b. Let us call this Game 0. The logic of the challenger in this game may be written as follows: Upon receiving m0 , m1 2 {0, 1}v from A, for some v L, do: b R {0, 1} G(s) s R {0, 1}` , r c r[0 . . v 1] mb send c to A. Game 0 is illustrated in Fig. 3.2. Let W0 be the event that ˆb = b in Game 0. By definition, we have SSadv⇤ [A, E] = Pr[W0 ]
1/2.
(3.4)
Next, we modify the challenger of Game 0, obtaining new game, called Game 1, which is exactly the same as Game 0, except that the challenger uses a truly random string in place of a pseudorandom string. The logic of the challenger in Game 1 is as follows: Upon receiving m0 , m1 2 {0, 1}v from A, for some v L, do: b R {0, 1} r R {0, 1}L c r[0 . . v 1] send c to A.
mb
49
b
R
s
R
r
Challenger
m0 , m1 2 {0, 1}
{0, 1}
(m0  = m1  = v)
{0, 1}
`
L
A
G(s)
c
r[0 . . v
1]
c
mb
ˆb 2 {0, 1}
Figure 3.2: Game 0 in the proof of Theorem 3.1
b
R
r
R
c
Challenger
m0 , m1 2 {0, 1}
{0, 1}
(m0  = m1  = v)
{0, 1}L
r[0 . . v
1]
L
A
mb
c ˆb 2 {0, 1}
Figure 3.3: Game 1 in the proof of Theorem 3.1
50
B m0 , m1 2 {0, 1} L (m0  = m1  = v) PRG Challenger for G
r 2 {0, 1}L
b
c
R
{0, 1}
r[0 . . v
1]
A
mb
c ˆb 2 {0, 1} (ˆb, b)
Figure 3.4: The PRG adversary B in the proof of Theorem 3.1 As usual, A outputs a bit ˆb at the end of this game. We have highlighted the changes from Game 0 in gray. Game 1 is illustrated in Fig. 3.3. Let W1 be the event that ˆb = b in Game 1. We claim that Pr[W1 ] = 1/2.
(3.5)
This is because in Game 1, the adversary is attacking the variable length onetime pad. In particular, it is easy to see that the adversary’s output ˆb and the challenger’s hidden bit b are independent. Finally, we show how to construct an efficient PRG adversary B that uses A as a subroutine, such that (3.6) Pr[W0 ] Pr[W1 ] = PRGadv[B, G]. This is actually quite straightforward. The logic of our new adversary B is illustrated in Fig. 3.4. Here, is defined as follows: ( 1 if x = y, (x, y) := (3.7) 0 if x 6= y. Also, the box labeled “PRG Challenger” is playing the role of the challenger in Attack Game 3.1 with respect to G. In words, adversary B, which is a PRG adversary designed to attack G (as in Attack Game 3.1), receives r 2 {0, 1}L from its PRG challenger, and then plays the role of challenger to A, as follows: Upon receiving m0 , m1 2 {0, 1}v from A, for some v L, do: b R {0, 1} c r[0 . . v 1] mb send c to A. 51
Finally, when A outputs a bit ˆb, B outputs the bit (ˆb, b). Let p0 be the probability that B outputs 1 when the PRG challenger is running Experiment 0 of Attack Game 3.1, and let p1 be the probability that B outputs 1 when the PRG challenger is running Experiment 1 of Attack Game 3.1. By definition, PRGadv[B, G] = p1 p0 . Moreover, if the PRG challenger is running Experiment 0, then adversary A is essentially playing our Game 0, and so p0 = Pr[W0 ], and if the PRG challenger is running Experiment 1, then A is essentially playing our Game 1, and so p1 = Pr[W1 ]. Equation (3.6) now follows immediately. Combining (3.4), (3.5), and (3.6), yields (3.3). 2 In the above theorem, we reduced the security of E to that of G by showing that if A is an efficient SS adversary that attacks E, then there exists an efficient PRG adversary B that attacks G, such that SSadv[A, E] 2 · PRGadv[B, G]. (Actually, we showed that equality holds, but that is not so important.) In the proof, we argued that if G is secure, then PRGadv[B, G] is negligible, hence by the above inequality, we conclude that SSadv[A, E] is also negligible. Since this holds for all efficient adversaries A, we conclude that E is semantically secure. Analogous to the discussion after the proof of Theorem 2.7, another way to structure the proof is by proving the contrapositive: indeed, if we assume that E is insecure, then there must be an efficient adversary A such that SSadv[A, E] is nonnegligible, and the reduction (and the above inequality) gives us an efficient adversary B such that PRGadv[B, G] is also nonnegligible. That is, if we can break E, we can also break G. While logically equivalent, such a proof has a di↵erent “feeling”: one starts with an adversary A that breaks E, and shows how to use A to construct a new adversary B that breaks G. The reader should notice that the proof of the above theorem follows the same basic pattern as our analysis of Internet roulette in Section 2.3.4. In both cases, we started with an attack game (Fig. 2.2 or Fig. 3.2) which we modified to obtain a new attack game (Fig. 2.3 or Fig. 3.3); in this new attack game, it was quite easy to compute the adversary’s advantage. Also, we used an appropriate security assumption to show that the di↵erence between the adversary’s advantages in the original and the modified games was negligible. This was done by exhibiting a new adversary (Fig. 2.4 or Fig. 3.4) that attacked the underlying cryptographic primitive (cipher or PRG) with an advantage equal to this di↵erence. Assuming the underlying primitive was secure, this di↵erence must be negligible; alternatively, one could argue the contrapositive: if this di↵erence were not negligible, the new adversary would “break” the underlying cryptographic primitive. This is a pattern that will be repeated and elaborated upon throughout this text. The reader is urged to study both of these analyses to make sure he or she completely understands what is going on.
3.3
Stream cipher limitations: attacks on the one time pad
Although stream ciphers are semantically secure they are highly brittle and become totally insecure if used incorrectly.
52
3.3.1
The twotime pad is insecure
A stream cipher is well equipped to encrypt a single message from Alice to Bob. Alice, however, may wish to send several messages to Bob. For simplicity suppose Alice wishes to encrypt two messages m1 and m2 . The naive solution is to encrypt both messages using the same stream cipher key s: m1 G(s) and c2 m2 G(s) (3.8) c1 A moments reflection shows that this construction is insecure in a very strong sense. An adversary who intercepts c1 and c2 can compute := c1
c2 = m1
G(s)
m2
G(s) = m1
m2
and obtain the xor of m1 and m2 . Not surprisingly, English text contains enough redundancy that given = m1 m2 the adversary can recover both m1 and m2 in the clear. Hence, the construction in (3.8) leaks the plaintexts after seeing only two sufficiently long ciphertexts. The construction in (3.8) is jokingly called the twotime pad. We just argued that the twotime pad is totally insecure. In particular, a stream cipher key should never be used to encrypt more than one message. Throughout the book we will see many examples where a onetime cipher is sufficient. For example, when choosing a new random key for every message as in Section 5.4.1. However, in settings where a single key is used multiple times, one should never use a stream cipher directly. We build multiuse ciphers in Chapter 5. Incorrectly reusing a stream cipher key is a common error in deployed systems. For example, a protocol called PPTP enables two parties A and B to send encrypted messages to one another. Microsoft’s implementation of PPTP in Windows NT uses a stream cipher called RC4. The original implementation encrypts messages from A to B using the same RC4 key as messages from B to A [95]. Consequently, by eavesdropping on two encrypted messages headed in opposite directions an attacker could recover the plaintext of both messages. Another amusing story about the twotime pad is relayed by Klehr [52] who describes in great detail how Russian spies in the US during World War II were sending messages back to Moscow, encrypted with the onetime pad. The system had a critical flaw, as explained by Klehr: During WWII the Soviet Union could not produce enough onetime pads . . . to keep up with the enormous demand . . . . So, they used a number of onetime pads twice, thinking it would not compromise their system. American counterintelligence during WWII collected all incoming and outgoing international cables. Beginning in 1946, it began an intensive e↵ort to break into the Soviet messages with the cooperation of the British and by . . . the Soviet error of using some onetime pads as twotime pads, was able, over the next 25 years, to break some 2900 messages, containing 5000 pages of the hundreds of thousands of messages that been sent between 1941 and 1946 (when the Soviets switched to a di↵erent system). The decryption e↵ort was codenamed project Venona. The Venona files are most famous for exposing Julius and Ethel Rosenberg and help give indisputable evidence of their involvement with the Soviet spy ring. Starting in 1995 all 3000 Venona decrypted messages were made public.
3.3.2
The onetime pad is malleable
Although semantic security ensures that an adversary cannot read the plaintext, it provides no guarantees for integrity. When using a stream cipher, an adversary can change a ciphertext and 53
the modification will never be detected by the decryptor. Even worse, let us show that by changing the ciphertext, the attacker can control how the decrypted plaintext will change. Suppose an attacker intercepts a ciphertext c := E(s, m) = m G(s). The attacker changes c to c0 := c for some of the attacker’s choice. Consequently, the decryptor receives the modified message ) G(s) = m . D(s, c0 ) = c0 G(s) = (c Hence, without knowledge of either m or s, the attacker was able to cause the decrypted message to become m for of the attacker’s choosing. We say that streamciphers are malleable since an attacker can cause predictable changes to the plaintext. We will construct ciphers that provide both privacy and integrity in Chapter 9. A simple example where malleability could help an attacker is an encrypted file system. To make things concrete, suppose Bob is a professor and that Alice and Molly are students. Bob’s students submit their homework by email, and then Bob stores these emails on a disk encrypted using a stream cipher. An email always starts with a standard header. Simplifying things a bit, we can assume that an email from, say, Alice, always starts with the characters From:Alice. Now suppose Molly is able to gain access to Bob’s disk and locate the encryption of the email from Alice containing her homework. Molly can e↵ectively steal Alice’s homework, as follows. She simply XORs the appropriate fivecharacter string into the ciphertext in positions 6 to 10, so as to change the header From:Alice to the header From:Molly. Molly makes this change by only operating on ciphertexts and without knowledge of Bob’s secret key. Bob will never know that the header was changed, and he will grade Alice’s homework, thinking it is Molly’s, and Molly will get the credit instead of Alice. Of course, for this attack to be e↵ective, Molly must somehow be able to find the email from Alice on Bob’s encrypted disk. However, in some implementations of encrypted file systems, file metadata (such as file names, modification times, etc) are not encrypted. Armed with this metadata, it may be straightforward for Molly to locate the encrypted email from Alice and carry out this attack.
3.4
Composing PRGs
In this section, we discuss two constructions that allow one to build new PRGs out of old PRGs. These constructions allow one to increase the size of the output space of the original PRG while at the same time preserving its security. Perhaps more important than the constructions themselves is the proof technique, which is called a hybrid argument. This proof technique is used pervasively throughout modern cryptography.
3.4.1
A parallel construction
Let G be a PRG defined over (S, R). Suppose that in some application, we want to use G many times. We want all the outputs of G to be computationally indistinguishable from random elements of R. If G is a secure PRG, and if the seeds are independently generated, then this will indeed be the case. We can model the use of many applications of G as a new PRG G0 . That is, we construct a new PRG G0 that applies G to n seeds, and concatenates the outputs. Thus, G0 is defined over (S n , Rn ), and for s1 , . . . , sn 2 R, G0 (s1 , . . . , sn ) := (G(s1 ), . . . , G(sn )). 54
We call G0 the nwise parallel composition of G. The value n is called a repetition parameter, and we require that it is a polybounded value. Theorem 3.2. If G is a secure PRG, then the nwise parallel composition G0 of G is also a secure PRG. In particular, for every PRG adversary A that attacks G0 as in Attack Game 3.1, there exists a PRG adversary B that attacks G as in Attack Game 3.1, where B is an elementary wrapper around A, such that PRGadv[A, G0 ] = n · PRGadv[B, G].
As a warm up, we first prove this theorem in the special case n = 2. Let A be an efficient PRG adversary that has advantage ✏ in attacking G0 in Attack Game 3.1. We want to show that ✏ is negligible, under the assumption that G is a secure PRG. To do this, let us define Game 0 to be Experiment 0 of Attack Game 3.1 with A and G0 . The challenger in this game works as follows: G(s1 ) s1 R S, r1 G(s2 ) s2 R S, r2 send (r1 , r2 ) to A. Let p0 denote the probability with which A outputs 1 in this game. Next, we define Game 1, which is played between A and a challenger that works as follows: r1 R R s2 R S, r2 G(s2 ) send (r1 , r2 ) to A. Note that Game 1 corresponds to neither Experiment 0 nor Experiment 1 of Attack Game 3.1; rather, it is a “hybrid” experiment corresponding to something in between Experiments 0 and 1. All we have done is replaced the pseudorandom value r1 in Game 0 by a truly random value (as highlighted). Intuitively, under the assumption that G is a secure PRG, the adversary A should not notice the di↵erence. To make this argument precise, let p1 be the probability that A outputs 1 in Game 1. Let 1 := p1 p0 . We claim that 1 is negligible, assuming that G is a secure PRG. Indeed, we can easily construct an efficient PRG adversary B1 whose advantage in attacking G in Attack Game 3.1 is precisely equal to 1 . The adversary B1 works as follows: Upon receiving r 2 R from its challenger, B1 plays the role of challenger to A, as follows: r r1 R S, r2 G(s2 ) s1 send (r1 , r2 ) to A. Finally, B1 outputs whatever A outputs. Observe that when B1 is in Experiment 0 of its attack game, it perfectly mimics the behavior of the challenger in Game 0, while in Experiment 1, it perfectly mimics the behavior of the challenger in Game 1. Thus, p0 is equal to the probability that B1 outputs 1 in Experiment 0 of Attack Game 3.1, while p1 is equal to the probability that B1 outputs 1 in Experiment 1 of Attack Game 3.1. Thus, B1 ’s advantage in attacking G is precisely p1 p0 , as claimed. Next, we define Game 2, which is played between A and a challenger that works as follows: 55
r1 R R r2 R R send (r1 , r2 ) to A. All we have done is replaced the pseudorandom value r2 in Game 1 by a truly random value (as highlighted). Let p2 be the probability that A outputs 1 in Game 2. Note that Game 2 corresponds to Experiment 1 of Attack Game 3.1 with A and G0 , and so p2 is equal to the probability that A outputs 1 in Experiment 1 of Attack Game 3.1 with respect to G0 . Let 2 := p2 p1 . By an argument similar to that above, it is easy to see that 2 is negligible, assuming that G is a secure PRG. Indeed, we can easily construct an efficient PRG adversary B2 whose advantage in Attack Game 3.1 with respect to G is precisely equal to 2 . The adversary B2 works as follows: Upon receiving r 2 R from its challenger, B2 plays the role of challenger to A, as follows: r1 R R r r2 send (r1 , r2 ) to A. Finally, B2 outputs whatever A outputs.
It should be clear that p1 is equal to the probability that B2 outputs 1 in Experiment 0 of Attack Game 3.1, while p2 is equal to the probability that B2 outputs 1 in Experiment 1 of Attack Game 3.1. Recalling that ✏ = PRGadv[A, G0 ], then from the above discussion, we have ✏ = p2
p0  = p2
p1 + p1
p0  p1
p0  + p2
p1  =
1
+
2.
Since both 1 and 2 are negligible, then so is ✏ (see Fact 2.6). That completes the proof that G0 is secure in the case n = 2. Before giving the proof in the general case, we give another proof in the case n = 2. While our first proof involved the construction of two adversaries B1 and B2 , our second proof combines these two adversaries into a single PRG adversary B that plays Attack Game 3.1 with respect to G, and which runs as follows: upon receiving r 2 R from its challenger, adversary B chooses ! 2 {1, 2} at random, and gives r to B! ; finally, B outputs whatever B! outputs.
Let W0 be the event that B outputs 1 in Experiment 0 of Attack Game 3.1, and W1 be the event that B outputs 1 in Experiment 1 of Attack Game 3.1. Conditioning on the events ! = 1 and ! = 2, we have Pr[W0 ] = Pr[W0  ! = 1] Pr[! = 1] + Pr[W0  ! = 2] Pr[! = 2] ✓ ◆ 1 = 2 Pr[W0  ! = 1] + Pr[W0  ! = 2] = 12 (p0 + p1 ).
Similarly, we have Pr[W1 ] = Pr[W1  ! = 1] Pr[! = 1] + Pr[W1  ! = 2] Pr[! = 2] ✓ ◆ 1 = 2 Pr[W1  ! = 1] + Pr[W1  ! = 2] = 12 (p1 + p2 ).
56
Therefore, if
is the advantage of B in Attack Game 3.1 with respect to G, we have = Pr[W1 ]
Thus, ✏ = 2 , and since
Pr[W0 ] =
1 2 (p1
+ p2 )
1 2 (p0
+ p1 ) = 12 p2
p0  = ✏/2.
is negligible, so is ✏ (see Fact 2.6).
Now, finally, we present the proof of Theorem 3.2 for general, polybounded n. Proof idea. We could try to extend the first strategy outlined above from n = 2 to arbitrary n. That is, we could construct a sequence of n + 1 games, starting with a challenger that produces a sequence (G(s1 ), . . . , G(sn )), of pseudorandom elements replacing elements one at a time with truly random elements of R, ending up with a sequence (r1 , . . . , rn ) of truly random elements of R. Intuitively, the adversary should not notice any of these replacements, since G is a secure PRG; however, proving this formally would require the construction of n di↵erent adversaries, each of which attacks G in a slightly di↵erent way. As it turns out, this leads to some annoying technical difficulties when n is not an absolute constant, but is simply polybounded; it is much more convenient to extend the second strategy outlined above, constructing a single adversary that attacks G “in one blow.” 2 Proof. Let A be an efficient PRG adversary that plays Attack Game 3.1 with respect to G0 . We first introduce a sequence of n + 1 hybrid games, called Hybrid 0, Hybrid 1, . . . , Hybrid n. For j = 0, 1, . . . , n, Hybrid j is a game played between A and a challenger that prepares a tuple of n values, the first j of which are truly random, and the remaining n j of which are pseudorandom outputs of G; that is, the challenger works as follows: r1
R
rj
R
sj+1 sn
R
R .. . R R
S, rj+1 G(sj+1 ) .. . S, rn G(sn )
send (r1 , . . . , rn ) to A.
As usual, A outputs 0 or 1 at the end of the game. Fig. 3.5 illustrates the values prepared by the challenger in each of these n + 1 games. Let pj denote the probability that A outputs 1 in Hybrid j. Note that p0 is also equal to the probability that A outputs 1 in Experiment 0 of Attack Game 3.1, while pn is equal to the probability that A outputs 1 in Experiment 1. Thus, we have PRGadv[A, G0 ] = pn
p0 .
(3.9)
We next define a PRG adversary B that plays Attack Game 3.1 with respect to G, and which works as follows: Upon receiving r 2 R from its challenger, B plays the role of challenger to A, as follows:
57
Hybrid 0: Hybrid 1: Hybrid 2: .. . Hybrid n 1: Hybrid n:
G(s1 ) r1 r1
G(s2 ) G(s2 ) r2
G(s3 ) G(s3 ) G(s3 )
··· ··· ···
G(sn ) G(sn ) G(sn )
r1 r1
r2 r2
r3 r3
··· ···
G(sn ) rn
Figure 3.5: Values prepared by challenger in Hybrids 0, 1, . . . , n. Each ri is a random element of R, and each si is a random element of S. R
! r1 r!
R
1
{1, . . . , n} R .. . R
r!
r
s!+1
R
sn
R
R
S, r!+1 G(s!+1 ) .. . S, rn G(sn )
send (r1 , . . . , rn ) to A.
Finally, B outputs whatever A outputs. Let W0 be the event that B outputs 1 in Experiment 0 of Attack Game 3.1, and W1 be the event that B outputs 1 in Experiment 1 of Attack Game 3.1. The key observation is this: conditioned on ! = j for every fixed j = 1, . . . , n, Experiment 0 of B’s attack game is equivalent to Hybrid j 1, while Experiment 1 of B’s attack game is equivalent to Hybrid j. Therefore, Pr[W0  ! = j] = pj
1
and
Pr[W1  ! = j] = pj .
So we have Pr[W0 ] =
n X j=1
n
n
j=1
j=1
n
n
j=1
j=1
1X 1X Pr[W0  ! = j] Pr[! = j] = Pr[W0  ! = j] = pj n n
1,
and similarly, Pr[W1 ] =
n X j=1
Pr[W1  ! = j] Pr[! = j] =
58
1X 1X Pr[W1  ! = j] = pj . n n
Finally, we have PRGadv[B, G] = Pr[W1 ] Pr[W0 ] n n 1X 1X = pj pj n n j=1
=
1 pn n
1
j=1
p0 ,
and combining this with (3.9), we have PRGadv[A, G0 ] = n · PRGadv[B, G]. Since we are assuming G is a secure PRG, it follows that PRGadv[B, G] is negligible, and since n is polybounded, it follows that PRGadv[A, G0 ] is negligible (see Fact 2.6). That proves the theorem. 2 Theorem 3.2 says that the security of a PRG degrades at most linearly in the number of times that we use it. One might ask if this bound is tight; that is, might security indeed degrade linearly in the number of uses? The answer is in fact “yes” (see Exercise 3.14).
3.4.2
A sequential construction: the BlumMicali method
We now present a sequential construction, invented by Blum and Micali, which uses a PRG that stretches just a little, and builds a PRG that stretches an arbitrary amount. Let G be a PRG defined over (S, R ⇥ S), for some finite sets S and R. For every polybounded value n 1, we can construct a new PRG G0 , defined over (S, Rn ⇥ S). For s 2 S, we let G0 (s) := s s0 for i 1 to n do G(si 1 ) (ri , si ) output (r1 , . . . , rn , sn ). We call G0 the nwise sequential composition of G. See Fig. 3.6 for a schematic description of G0 for n = 3. We shall prove below in Theorem 3.3 that if G is a secure PRG, then so is G0 . As a special case of this construction, suppose G is a PRG defined over ({0, 1}` , {0, 1}t+` ), for some positive integers ` and t; that is, G stretches `bit strings to (t + `)bit strings. We can naturally view the output space of G as {0, 1}t ⇥ {0, 1}` , and applying the above construction, and interpreting outputs as bit strings, we get a PRG G0 that stretches `bit strings to (nt + `)bit strings. Theorem 3.3. If G is a secure PRG, then the nwise sequential composition G0 of G is also a secure PRG. In particular, for every PRG adversary A that plays Attack Game 3.1 with respect to G0 , there exists a PRG adversary B that plays Attack Game 3.1 with respect to G, where B is an elementary wrapper around A, such that PRGadv[A, G0 ] = n · PRGadv[B, G].
59
s
s2
s1 G
G
G
r2
r1
r3
s3
Figure 3.6: The sequential construction for n = 3 Proof idea. The proof of this is a hybrid argument that is very similar in spirit to the proof of Theorem 3.2. The intuition behind the proof is as follows: Consider a PRG adversary A who receives the (r1 , . . . , rn , sn ) Experiment 0 of Attack Game 3.1. Since s = s0 is random and G is a secure PRG, we may replace (r1 , s1 ) by a completely random element of R ⇥ S, and the probability that A outputs 1 in this new, hybrid game should change by only a negligible amount. Now, since s1 is random (and again, since G is a secure PRG), we may replace (r2 , s2 ) by a completely random element of R ⇥ S, and the probability that A outputs 1 in this second hybrid game should again change by only a negligible amount. Continuing in this way, we may incrementally replace (r3 , s3 ) through (rn , sn ) by random elements of R ⇥ S, and the probability that A outputs 1 should change by only a negligible amount after making all these changes (assuming n is polybounded). However, at this point, A outputs 1 with the same probability with which he would output 1 in Experiment 1 in Attack Game 3.1, and therefore, this probability is negligibly close to the probability that A outputs 1 in Experiment 0 of Attack Game 3.1. That is the idea; however, just as in the proof of Theorem 3.2, for technical reasons, we design a single PRG adversary that attacks G. 2 Proof. Let A be a PRG adversary that plays Attack Game 3.1 with respect to G0 . We first introduce a sequence of n + 1 hybrid games, called Hybrid 0, Hybrid 1, . . . , Hybrid n. For j = 0, 1, . . . , n, we define Hybrid j to be the game played between A and the following challenger: r1
R
rj
R
sj
R
R .. . R S
(rj+1 , sj+1 ) .. . (rn , sn )
G(sj )
G(sn
1)
send (r1 , . . . , rn , sn ) to A. As usual, A outputs 0 or 1 at the end of the game. See Fig. 3.7 for a schematic description of how these challengers work in the case n = 3. Let pj denote the probability that A outputs 1 in Hybrid j. Note that p0 is also equal to the probability that A outputs 1 in Experiment 0 of 60
Attack Game 3.1, while pn is equal to the probability that A outputs 1 in Experiment 1 of Attack Game 3.1. Thus, we have (3.10) PRGadv[A, G0 ] = pn p0 . We next define a PRG adversary B that plays Attack Game 3.1 with respect to G, and which works as follows: Upon receiving (r, s) 2 R ⇥ S from its challenger, B plays the role of challenger to A, as follows: ! r1 r!
R R
1
{1, . . . , n} R .. . R
(r! , s! )
R
(r, s)
(r!+1 , s!+1 ) .. . (rn , sn )
G(s! )
G(sn
1)
send (r1 , . . . , rn , sn ) to A. Finally, B outputs whatever A outputs. Let W0 be the event that B outputs 1 in Experiment 0 of Attack Game 3.1, and W1 be the event that B outputs 1 in Experiment 1 of Attack Game 3.1. The key observation is this: conditioned on ! = j for every fixed j = 1, . . . , n, Experiment 0 of B’s attack game is equivalent to Hybrid j 1, while Experiment 1 of B’s attack game is equivalent to Hybrid j. Therefore, Pr[W0  ! = j] = pj
1
and
Pr[W1  ! = j] = pj .
The remainder of the proof is a simple calculation that is identical to that in the last paragraph of the proof of Theorem 3.2. 2 One criteria for evaluating a PRG is its expansion rate: a PRG that stretches an nbit seed to an mbit output has expansion rate of m/n; more generally, if the seed space is S and the output space is R, we would define the expansion rate as logR/ logS. The sequential composition achieves a better expansion rate than the parallel composition. However, it su↵ers from the drawback that it cannot be parallelized. In fact, we can obtain the best of both worlds: a large expansion rate with a highly parallelizable construction (see Section 4.4.4).
3.4.3
Mathematical details
There are some subtle points in the proofs of Theorems 3.2 and 3.3 that merit discussion. First, in both constructions, the underlying PRG G may have system parameters. That is, there may be a probabilistic algorithm that takes as input the security parameter , and outputs a system parameter ⇤. Recall that a system parameter is public data that fully instantiates the 61
Hybrid 0 S
G
G
G
r2
r1
r3
s3
r3
s3
r3
s3
r3
s3
Hybrid 1 S
G
G
R
r2
r1 Hybrid 2
S
R
G
R
r2
r1 Hybrid 3 R
S
R
R
r2
r1
Figure 3.7: The challenger’s computation in the hybrid games for n = 3. The circles indicate randomly generated elements of S or R, as indicated by the label.
62
scheme (in this case, it might define the seed and output spaces). For both the parallel and sequential constructions, one could use the same system parameter for all n instances of G; in fact, for the sequential construction, this is necessary to ensure that outputs from one round may be used as inputs in the next round. The proofs of these security theorems are perfectly valid if the same system parameter is used for all instances of G, or if di↵erent system parameters are used. Second, we briefly discuss a rather esoteric point regarding hybrid arguments. To make things concrete, we focus attention on the proof of Theorem 3.2 (although analogous remarks apply to the proof of Theorem 3.3, or any other hybrid argument). In proving this theorem, we ultimately want to show that if there is an efficient adversary A that breaks G0 , then there is an efficient adversary that breaks G. Suppose that A is an efficient adversary that breaks G0 , so that its advantage ✏( ) (which we write here explicitly as a function of the security parameter ) with respect to G0 is not negligible. This means that there exists a constant c such that ✏( ) 1/ c for infinitely many . Now, in the discussion preceding the proof of Theorem 3.2, we considered the special case n = 2, and showed that there exist efficient adversaries B1 and B2 , such that ✏( ) 1 ( ) + 2 ( ) for all , where j ( ) is the advantage of Bj with respect to G. It follows that either 1 ( ) 1/2 c infinitely often, or 2 ( ) 1/2 c infinitely often. So we may conclude that either B1 breaks G or B2 breaks G (or possibly both). Thus, there exists an efficient adversary that breaks G: it is either B1 or B2 , which one we do not say (and we do not have to). However, whichever one it is, it is a fixed adversary that is defined uniformly for all ; that is, it is a fixed machine that takes as input. This argument is perfectly valid, and extends to every constant n: we would construct n adversaries B1 , . . . , Bn , and argue that for some j = 1, . . . , n, adversary Bj must have advantage 1/n c infinitely often, and thus break G. However, this argument does not extend to the case where n is a function of , which we now write explicitly as n( ). The problem is not that 1/(n( ) c ) is perhaps too small (it is not). The problem is quite subtle, so before we discuss it, let us first review the (valid) proof that we did give. For each , we defined a sequence of n( ) + 1 hybrid games, so that for each , we actually get a di↵erent sequence of games. Indeed, we cannot speak of a single, finite sequence of games that works for all , since n( ) ! 1. Nevertheless, we explicitly constructed a fixed adversary B that is defined uniformly for all ; that is, B is a fixed machine that takes as input. The sequence of hybrid games that we define for each is a mathematical object for which we make no claims as to its computability — it is simply a convenient device used in the analysis of B. Hopefully by now the reader has at least a hint of the problem that arises if we attempt to generalize the argument for constant n to a function n( ). First of all, it is not even clear what it means to talk about n( ) adversaries B1 , . . . , Bn( ) : our adversaries our supposed to be fixed machines that take as input, and the machines themselves should not depend on . Such linguistic confusion aside, our proof for the constant case only shows that there exists an “adversary” that for infinitely many values of somehow knows the “right” value of j = j( ) to use in the (n( ) + 1)game hybrid argument — no single, constant value of j necessarily works for infinitely many . One can actually make sense of this type of argument if one uses a nonuniform model of computation, but we shall not take this approach in this text. All of these problems simply go away when we use a hybrid argument that constructs a single adversary B, as we did in the proofs of Theorems 3.2 and 3.3. However, we reiterate that the original analysis we did in the where n = 2, or its natural extension to every constant n, is perfectly valid. In that case, we construct a single, fixed sequence of n + 1 games, with each individual game uniformly defined for all (just as our attack games are in our security definitions), as well as a
63
finite collection of adversaries, each of which is a fixed machine. We reiterate this because in the sequel we shall often be constructing proofs that involve finite sequences of games like this (indeed, the proof of Theorem 3.1 was of this type). In such cases, each game will be uniformly defined for all , and will be denoted Game 0, Game 1, etc. In contrast, when we make a hybrid argument that uses nonuniform sequences of games, we shall denote these games Hybrid 0, Hybrid 1, etc., so as to avoid any possible confusion.
3.5
The next bit test
Let G be a PRG defined over ({0, 1}` , {0, 1}L ), so that it stretches `bit strings to Lbit strings. There are a number of ways an adversary might be able to distinguish a pseudorandom output of G from a truly random bit string. Indeed, suppose that an efficient adversary were able to compute, say, the last bit of G’s output, given the first L 1 bits of G’s output. Intuitively, the existence of such an adversary would imply that G is insecure, since given the first L 1 bits of a truly random Lbit string, one has at best a 5050 chance of guessing the last bit. It turns out that an interesting converse, of sorts, is also true. We shall formally define the notion of unpredictability for a PRG, which essentially says that given the first i bits of G’s output, it is hard to predict the next bit (i.e., the (i + 1)st bit) with probability significantly better that 1/2 (here, i is an adversarially chosen index). We shall then prove that unpredictability and security are equivalent. The fact that security implies unpredictability is fairly obvious: the ability to e↵ectively predict the next bit in the pseudorandom output string immediately gives an e↵ective statistical test. However, the fact that unpredictability implies security is quite interesting (and requires more e↵ort to prove): it says that if there is any e↵ective statistical test at all, then there is in fact an e↵ective method for predicting the next bit in a pseudorandom output string. Attack Game 3.2 (Unpredictable PRG). For a given PRG G, defined over (S, {0, 1}L ), and a given adversary A, the attack game proceeds as follows: • The adversary sends an index i, with 0 i L • The challenger computes and sends r[0 . . i
s
R
S, r
1, to the challenger. G(s)
1] to the adversary.
• The adversary outputs g 2 {0, 1}. We say that A wins if r[i] = g, and we define A’s advantage Predadv[A, G] to be Pr[A wins] 1/2. 2 Definition 3.3 (Unpredictable PRG). A PRG G is unpredictable if the value Predadv[A, G] is negligible for all efficient adversaries A. We begin by showing the security implies unpredictability. Theorem 3.4. Let G be a PRG, defined over (S, {0, 1}L ). If G is secure, then G is unpredictable.
64
In particular, for every adversary A breaking the unpredictability of G, as in Attack Game 3.2, there exists an adversary B breaking the security G as in Attack Game 3.1, where B is an elementary wrapper around A, such that Predadv[A, G] = PRGadv[B, G].
Proof. Let A be an adversary breaking the predictability of G, and let i denote the index chosen by A. Also, suppose A wins Attack Game 3.2 with probability 1/2 + ✏, so that Predadv[A, G] = ✏. We build an adversary B breaking the security of G, using A as a subroutine, as follows: Upon receiving r 2 {0, 1}L from its challenger, B does the following: • B gives r[0 . . i
1] to A, obtaining A’s output g 2 {0, 1};
• if r[i] = g, then output 1, and otherwise, output 0.
For b = 0, 1, let Wb be the event that B outputs 1 in Experiment b of Attack Game 3.1. In Experiment 0, r is a pseudorandom output of G, and W0 occurs if and only if r[i] = g, and so by definition Pr[W0 ] = 1/2 + ✏. In Experiment 1, r is a truly random bit string, but again, W1 occurs if and only if r[i] = g; in this case, however, as random variables, the values of r[i] and g are independent, and so Pr[W1 ] = 1/2. It follows that PRGadv[B, G] = Pr[W1 ]
Pr[W0 ] = ✏ = Predadv[A, G].
2
The more interesting, and more challenging, task is to show that unpredictability implies security. Before getting into all the details of the proof, we sketch the high level ideas. First, we shall employ a hybrid argument, which will essentially allow us to argue that if A is an efficient adversary that can e↵ectively distinguish a pseudorandom Lbit string from a random Lbit string, then we can construct an efficient adversary B that can e↵ectively distinguish x1 · · · xj xj+1 from x1 · · · xj r, where j is a randomly chosen index, x1 , . . . , xL is the pseudorandom output, and r is a random bit. Thus, adversary B can distinguish the pseudorandom bit xj+1 from the random bit rj+1 , given the “side information” x1 , . . . , xj . We want to turn B’s distinguishing advantage into a predicting advantage. The rough idea is this: given x1 , . . . , xj , we feed B the string x1 , . . . , xj r for a randomly chosen bit r; if B outputs 1, our prediction for xj+1 is r; otherwise, or prediction for xj+1 is r¯ (the complement of r). That this prediction strategy works is justified by the following general result, which we call the distinguisher/predictor lemma. The general setup is as follows. We have: • a random variable X, which corresponds to the “side information” x1 , . . . , xj above, as well as any random coins used by the adversary B; 65
• a 0/1valued random variable B, which corresponds to xj+1 above, and which may be correlated with X; • a 0/1valued random variable R, which corresponds to r above, and which is independent of (X, B); • a function d, which corresponds to B’s strategy, so that B’s distinguishing advantage is equal to ✏, where ✏ = Pr[d(X, B) = 1] Pr[d(X, R) = 1].
The lemma says that if we define B0 using the predicting strategy outlined above, namely B0 = R if d(X, R) = 1, and B0 = R otherwise, then the probability that the prediction B0 is equal to the actual value B is precisely 1/2 + ✏. Here is the precise statement of the lemma: Lemma 3.5 (Distinguisher/predictor lemma). Let X be a random variable taking values in some set S, and let B and R be a 0/1valued random variables, where R is uniformly distributed over {0, 1} and is independent of (X, B). Let d : S ⇥ {0, 1} ! {0, 1} be an arbitrary function, and let ✏ := Pr[d(X, B) = 1] Pr[d(X, R) = 1]. Define the random variable B0 as follows: ( B0 :=
R R
if d(X, R) = 1; otherwise.
Then Pr[B0 = B] = 1/2 + ✏. Proof. We calculate Pr[B0 = B], conditioning on the events B = R and B = R: Pr[B0 = B] = Pr[B0 = B  B = R] Pr[B = R] + Pr[B0 = B  B = R] Pr[B = R] 1 1 = Pr[d(X, R) = 1  B = R] + Pr[d(X, R) = 0  B = R] 2 2 ⌘ 1⇣ = Pr[d(X, R) = 1  B = R] + (1 Pr[d(X, R) = 1  B = R)] 2 1 1 = + (↵ ), 2 2 where ↵ := Pr[d(X, R) = 1  B = R] and
By independence, we have
:= Pr[d(X, R) = 1  B = R].
↵ = Pr[d(X, R) = 1  B = R] = Pr[d(X, B) = 1  B = R] = Pr[d(X, B) = 1]. To see the last equality, the result of Exercise 3.25 may be helpful. We thus calculate that ✏ = Pr[d(X, B) = 1] Pr[d(X, R) = 1] ⌘ ⇣ =↵ Pr[d(X, R) = 1  B = R] Pr[B = R] + Pr[d(X, R) = 1  B = R] Pr[B = R] =↵
1 = (↵ 2
1 (↵ + ) 2 ),
66
which proves the lemma. 2 Theorem 3.6. Let G be a PRG, defined over (S, {0, 1}L ). If G is unpredictable, then G is secure. In particular, for every adversary A breaking the security of G as in Attack Game 3.1, there exists an adversary B, breaking the unpredictability of G as in Attack Game 3.2, where B is an elementary wrapper around A, such that PRGadv[A, G] = L · Predadv[B, G].
Proof. Let A attack G as in Attack Game 3.1. Using A, we build a predictor B, which attacks G as in Attack Game 3.2, and works as follows: • Choose ! 2 {1, . . . , L} at random. • Send L
! to the challenger, obtaining a string x 2 {0, 1}L
!.
• Generate ! random bits r1 , . . . , r! , and give the Lbit string x k r1 · · · r! to A. • If A outputs 1, then output r1 ; otherwise, output r1 . To analyze B, we consider L + 1 hybrid games, called Hybrid 0, Hybrid 1, . . . , Hybrid L. For j = 0, . . . , L, we define Hybrid j to be the game played between A and a challenger that generates a bit string r consisting of L j pseudorandom bits, followed by j truly random bits; that is, the challenger chooses s 2 S and t 2 {0, 1}j at random, and sends A the bit string r := G(s)[0 . . L
1] k t.
j
As usual, A outputs 0 or 1 at the end of the game, and we define pj to be the probability that A outputs 1 in Hybrid j. Note that p0 is the probability that A outputs 1 in Experiment 0 of Attack Game 3.1, while pL is the probability that A outputs 1 in Experiment 1 of Attack Game 3.1. Let W be the event that B wins in Attack Game 3.2 (that is, correctly predicts the next bit). Then we have Pr[W ] =
L X j=1
Pr[W  ! = j] Pr[! = j] L
1X = Pr[W  ! = j] L j=1
1 X ⇣1 = + pj L 2 L
1
j=1
=
1 1 + (p0 2 L
pL ),
and the theorem follows. 2
67
pj
⌘
(by Lemma 3.5)
3.6
Case study: the Salsa and ChaCha PRGs
There are many ways to build PRGs and stream ciphers in practice. One approach builds PRGs using the BlumMicali paradigm discussed in Section 3.4.2. Another approach, discussed more generally in the Chapter 5, builds them from a more versatile primitive called a pseudorandom function in counter mode. We start with a construction that uses this latter approach. Salsa20/12 and Salsa20/20 are fast stream ciphers designed by Dan Burnstein in 2005. Salsa20/12 is one of four Profile 1 stream ciphers selected for the eStream portfolio of stream ciphers. eStream is a project that identifies fast and secure stream ciphers that are appropriate for practical use. Variants of Salsa20/12 and Salsa20/20, called ChaCha12 and ChaCha20 respectively, were proposed by Bernstein in 2008. These stream ciphers have been incorporated into several widely deployed protocols such as TLS and SSH. Let us briefly describe the PRGs underlying the Salsa and ChaCha stream cipher families. These PRGs take as input a 256bit seed and a 64bit nonce. For now we ignore the nonce and simply set it to 0. We discuss the purpose of the nonce at the end of this section. The Salsa and ChaCha PRGs follow the same high level structure shown in Fig. 3.8. They make use of two components: • A padding function denoted pad(s, j, 0) that combines a 256bit seed s with a 64bit counter j to form a 512bit block. The third input, a 64bit nonce, is always set to 0 for now. • A fixed public permutation ⇡ : {0, 1}512 ! {0, 1}512 . These components are used to output L < 264 pseudorandom blocks, each 512 bits long, using the following algorithm (Fig. 3.8): input: seed s 2 {0, 1}256
1. 2. 3.
for j
0 to L 1 hj pad(s, j, 0) 2 {0, 1}512 rj ⇡(hj ) hj
4.
output (r0 , . . . , rL
1 ).
The final PRG output is 512 · L bits long. We note that in Salsa and ChaCha the XOR on line 3 is a slightly more complicated operation: the 512bit operands hj and ⇡(hj ) are split into 16 words each 32bits long and then added wordwise mod 232 . The design of Salsa and ChaCha is highly parallelizable and can take advantage of multiple processor cores to speedup encryption. Moreover, it enables random access to output blocks: output block number j can be computed without having to first compute all previous blocks. Generators based on the BlumMicali paradigm do not have these properties. We analyze the security of the Salsa and ChaCha design in Exercise 4.23 in the next chapter, after we develop a few more tools. The details. We briefly describe the padding function pad(s, j, n) and the permutation ⇡ used in ChaCha20. The padding function takes as input a 256bit seed s0 , . . . , s7 2 {0, 1}32 , a 64bit counter j0 , j1 2 {0, 1}32 , and 64bit nonce n0 , n1 2 {0, 1}32 . It outputs a 512bit block denoted
68
seed" 256"bits"
pad(" ,"0","0)"
pad(" ,"1","0)"
pad(" ,"2","0)"
π"
π"
512"bits"
π" !
!
!
output"block"#0"
output"block"#1"
output"block"#2"
512"bits"
512"bits"
512"bits"
!
!
Figure 3.8: A schematic of the Salsa and ChaCha PRGs x0 , . . . , x15 2 {0, 1}32 . The output is 0 x0 x 1 B x4 x5 B @ x8 x9 x12 x13
arranged in a 4 ⇥ 4 matrix 1 0 c0 c 1 x2 x3 C B x6 x7 C B s0 s1 @ s4 s5 x10 x11 A x14 x15 j0 j1
of 32bit words as follows: 1 c2 c3 s2 s3 C C (3.11) s6 s7 A n0 n1
where c0 , c1 , c2 , c3 are fixed 32bit constants. The permutation ⇡ : {0, 1}512 ! {0, 1}512 is constructed by iterating a simpler permutation a fixed number of times. The 512bit input to ⇡ is treated as a 4 ⇥ 4 array of 32bit words denoted by x0 , . . . , x15 . In ChaCha20 the function ⇡ is implemented by repeating the following sequence of steps ten times: QuarterRound(x0 , x4 , x8 , x12 ), QuarterRound(x1 , x5 , x9 , x13 ), QuarterRound(x2 , x6 , x10 , x14 ), QuarterRound(x3 , x7 , x11 , x15 ), QuarterRound(x0 , x5 , x10 , x15 ), QuarterRound(x1 , x6 , x11 , x12 ), QuarterRound(x2 , x7 , x8 , x13 ), QuarterRound(x3 , x4 , x9 , x14 ) where QuarterRound(a, b, c, d) is defined as the following sequence of steps written as C code: a c a c
+= += += +=
b; d; b; d;
d b d b
^= ^= ^= ^=
a; c; a; c;
d b d b
0 ctr 0 upon receiving a query x = (a1 , . . . , an ) 2 {0, 1}` from A do: if n < ! then y R S else u (a1 , . . . , a! 1 ), d a! , v (a!+1 , . . . , n) if u 2 / Domain(Map) then ctr ctr + 1, Map[u] ctr ⇤ p Map[u], y G (rpd , v) send y to A.
Finally, B 0 outputs whatever A outputs. For b = 0, 1, let Wb be the event that B 0 outputs 1 in Experiment b of Attack Game 4.2 with respect to G0 . It is not too hard to see that for any fixed j = 1, . . . , `, we have Pr[W0  ! = j] = pj
1
and
Pr[W1  ! = j] = pj .
Indeed, condition on ! = j for fixed j, and consider how B 0 labels nodes in the evaluation tree. At the line marked (⇤), B 0 assigns random labels to all nodes in the evaluation tree at levels 0 through j 1, and the assumption that A never makes the same query twice guarantees that these labels are consistent (the same node does not receive two di↵erent labels at di↵erent times). Now, on the one hand, when B 0 is in Experiment 1 of its attack game, it e↵ectively assigns random labels to nodes at level j as well, and the lookup table ensures that this is done consistently. On the other hand, when B 0 is in Experiment 0 of its attack game, it e↵ectively assigns pseudorandom labels to nodes at level j, which is the same as assigning random labels to the parents of these nodes at level 150
j 1; the prefixfreeness assumption ensures that none of these parent nodes are inconsistently assigned random labels at the line marked (⇤). The rest of the proof goes through as in the proof of Theorem 4.10. 2
4.7
The ideal cipher model
Block ciphers are used in a variety of cryptographic constructions. Sometimes it is impossible or difficult to prove a security theorem for some of these constructions under standard security assumptions. In these situations, a heuristic technique — called the ideal cipher model — is sometimes employed. Roughly speaking, in this model, the security analysis is done by treating the block cipher as if it were a family of random permutations. If E = (E, D) is a block cipher defined over (K, X ), then the family of random permutations is {⇧k }k 2K , where each ⇧k is a truly random permutation on X , and the ⇧k ’s collectively are mutually independent. These random permutations are much too large to write down and cannot be used in a real construction. Rather, they are used to model a construction based on a real block cipher, to obtain a heuristic security argument for a given construction. We stress the heuristic nature of the ideal cipher model: while a proof of security in this model is better than nothing, it does not rule out an attack by an adversary that exploits the design of a particular block cipher, even one that is secure in the sense of Definition 4.1.
4.7.1
Formal definitions
Suppose we have some type of cryptographic scheme S whose implementation makes use of a block cipher E = (E, D) defined over (K, X ). Moreover, suppose the scheme S evaluates E at various inputs (k , a ) 2 K ⇥ X , and D at various inputs (k , b ) 2 K ⇥ X , but does not look at the internal implementation of E. In this case, we say that S uses E as an oracle. We wish to analyze the security of S. Let us assume that whatever security property we are interested in, say “property X,” is modeled (as usual) as a game between a challenger (specific to property X) and an arbitrary adversary A. Presumably, in responding to certain queries, the challenger computes various functions associated with the scheme S, and these functions may in turn require the evaluation of E and/or D at certain points. This game defines an advantage Xadv[A, S], and security with respect to property X means that this advantage should be negligible for all efficient adversaries A. If we wish to analyze S in the ideal cipher model, then the attack game defining security is modified so that E is e↵ectively replaced by a family of random permutations {⇧k }k 2K , as described above, to which both the adversary and the challenger have oracle access. More precisely, the game is modified as follows. • At the beginning of the game, the challenger chooses ⇧k 2 Perms[K] at random, for each k 2 K. • In addition to its standard queries, the adversary A may submit ideal cipher queries. There are two types of queries: ⇧queries and ⇧ 1 queries. – For a ⇧query, the adversary submits a pair (k , a ) 2 K ⇥ X , to which the challenger responds with ⇧k (a ).
151
– For a ⇧ 1 query, the adversary submits a pair (k , b ) 2 K ⇥ X , to which the challenger responds with ⇧k 1 (b ). The adversary may make any number of ideal cipher queries, arbitrarily interleaved with standard queries. • In processing standard queries, the challenger performs its computations using ⇧k (a ) in place of E(k , a ) and ⇧k 1 (b ) in place of D(k , b ). The adversary’s advantage is defined using the same rule as before, but is denoted Xic adv[A, S] to emphasize that this is an advantage in the ideal cipher model. Security in the ideal cipher model means that Xic adv[A, S] should be negligible for all efficient adversaries A. It is important to understand the role of the ideal cipher queries. Essentially, they model the ability of an adversary to make “o✏ine” evaluations of E and D. Ideal permutation model. Some constructions, like EvenMansour (discussed below), make use of a permutation ⇡ : X ! X , rather than a block cipher. In the security analysis, one might heuristically model ⇡ as a random permutation ⇧, to which all parties in the attack game have oracle access to ⇧ and ⇧ 1 . We call this the ideal permutation model. One can view this as a special case of the ideal cipher model by simply defining ⇧ = ⇧k 0 for some fixed, publicly available key k 0 2 K.
4.7.2
Exhaustive search in the ideal cipher model
Let (E, D) be a block cipher defined over (K, X ) and let k be some random secret key in K. Suppose an adversary is able to intercept a small number of input/output pairs (xi , yi ) generated using k: yi = E(k, xi )
for all i = 1, . . . , Q.
The adversary can now recover k by trying all possible keys in k 2 K until a key k satisfying yi = E(k , xi ) for all i = 1, . . . , Q is found. For block ciphers used in practice it is likely that this k is equal to the secret key k used to generate the given pairs. This exhaustive search over the key space recovers the blockcipher secretkey in time O(K) using a small number of input/output pairs. We analyze the number of input/output pairs needed to mount a successful attack in Theorem 4.12 below. Exhaustive search is the simplest example of a keyrecovery attack. Since we will present a number of keyrecovery attacks, let us first define the keyrecovery attack game in more detail. We will primarily use the keyrecovery game as means of presenting attacks. Attack Game 4.4 (keyrecovery). For a given block cipher E = (E, D), defined over (K, X ), and for a given adversary A, define the following game: • The challenger picks a random k
R
K.
• A queries the challenger several times. For i = 1, 2, . . . , the ith query consists of a message xi 2 M. The challenger, given xi , computes yi R E(k, xi ), and gives yi to A. • Eventually A outputs an candidate key k 2 K. 152
We say that A wins the game if k = k. We let KRadv[A, E] denote the probability that A wins the game. 2 The keyrecovery game extends naturally to the ideal cipher model, where E(k , a ) = ⇧k (a ) and D(k , b ) = ⇧k 1 (b ), and {⇧k }k 2K is a family of independent random permutations. In this model, we allow the adversary to make arbitrary ⇧ and ⇧ 1 queries, in addition to its standard queries to E(k, ·). We let KRic adv[A, E] denote the adversary’s keyrecovery advantage when E is an ideal cipher. It is worth noting that security against keyrecovery attacks does not imply security in the sense of indistinguishability (Definition 4.1). The simplest example is the constant block cipher E(k, x) = x for which keyrecovery is not possible (the adversary obtains no information about k), but the block cipher is easily distinguished from a random permutation. Exhaustive search. The following theorem bounds the number of input/output pairs needed for exhaustive search, assuming the cipher is an ideal cipher. For realworld parameters, taking Q = 3 in the theorem is often sufficient to ensure success. Theorem 4.12. Let E = (E, D) be a block cipher defined over (K, X ). Then there exists an adversary AEX that plays Attack Game 4.4 with respect to E, modeled as an ideal cipher, making Q standard queries and QK ideal cipher queries, such that KRic adv[AEX , E]
(1
✏)
where
✏ :=
K (X  Q)Q
(4.34)
Proof. In the ideal cipher model, we are modeling the block cipher E = (E, D) as a family {⇧k }k 2K of random permutations on X . In Attack Game 4.4, the challenger chooses k 2 K at random. An adversary may make standard queries to obtain the value E(k, x) = ⇧k (x) at points x 2 X of his choosing. An adversary may also make ideal cipher queries, obtaining the values ⇧k (a ) and ⇧k 1 (b ) for points k 2 K and a , b 2 X of his choosing. These ideal cipher queries correspond to “o✏ine” evaluations of E and D. Our adversary AEX works as follows: let {x1 , . . . , xQ } be an arbitrary set of distinct messages in X for i = 1, . . . , Q do: make a standard query to obtain yi := E(k, xi ) = ⇧k (xi ) for each k 2 K do: for i = 1, . . . , Q do: make an ideal cipher query to obtain b i := ⇧k (xi ) if yi = b i for all i = 1, . . . , Q then output k and terminate Let k be the challenger’s secretkey. We show that AEX outputs k with probability at least 1 ✏, with ✏ defined as in (4.34). Since AEX tries all keys, this amounts to showing that the probability that there is more than one key consistent with the given (xi , yi ) pairs is at most ✏. We shall show that this holds for every possible choice of k, so for the remainder of the proof, we shall view k as fixed. We shall also view x1 , . . . , xQ as fixed, so all the probabilities are with respect to the random permutations ⇧k for k 2 K. 153
For each k 2 K, let Wk be the event that yi = ⇧k (xi ) for all i = 1, . . . , Q. Note that by definition, Wk occurs with probability 1. Let W be the event that Wk occurs for some k 6= k. We want to show that Pr[W ] ✏. Fix k 6= k. Since the permutation ⇧k is chosen independently of the permutation ⇧k , we know that ✓ ◆Q 1 1 1 1 Pr[Wk ] = · ··· X  X  1 X  Q + 1 X  Q As this holds for all k 6= k, the result follows from the union bound. 2 Security of the 3E construction The attack presented in Theorem 4.2 works equally well against the 3E construction. The size of the key space is K3 , but one obtains a “meet in the middle” key recovery algorithm that runs in time O K2 ·Q . For TripleDES this algorithm requires more than 22·56 evaluations of TripleDES, which is far beyond our computing power. One wonders whether better attacks against 3E exist. When E is an ideal cipher we can prove a lower bound on the amount of work needed to distinguish 3E from a random permutation. Theorem 4.13. Let E = (E, D) be an ideal block cipher defined over (K, X ), and consider an attack against the 3E construction in the ideal cipher model. If A is an adversary that makes at most Q queries (including both standard and ideal cipher queries) in the ideal cipher variant of Attack Game 4.1, then BCic adv[A, 3E] C1 L
Q2/3 1 Q2 , + C + C3 2 3 2/3 1/3 K K K X 
where L := max(K/X , log2 X ), and C1 , C2 , C3 are constants (that do not depend on A or E). The statement of the theorem is easier to understand if we assume that K X , as is the case with DES. In this case, the bound can be restated as BCic adv[A, 3E] C log2 X 
Q2 , K3
for a constant C. Ignoring the log X term, this says that an adversary must make roughly K1.5 queries to obtain a significant advantage (say, 1/4). Compare this to the meetinthemiddle attack. To achieve a significant advantage, that adversary must make roughly K2 queries. Thus, meetinthemiddle attack may not be the most powerful attack. To conclude our discussion of TripleDES, we note that the 3E construction does not always strengthen the cipher. For example, if E = (E, D) is such that the set of K permutations {E(k , ·) : k 2 K} is a group, then 3E would be no more secure than E. Indeed, in this case ⇡ := E3 ((k1 , k2 , k3 ), ·) is identical to E(k, ·) for some k 2 K. Consequently, distinguishing 3E from a random permutation is no harder than doing so for E. Of course, block ciphers used in practice are not groups (as far as we know).
154
4.7.3
The EvenMansour block cipher and the EX construction
Let X = {0, 1}n . Let ⇡ : X ! X be a permutation and let ⇡ 1 be its inverse function. Even and Mansour defined the following simple block cipher EEM = (E, D) defined over (X 2 , X ): E (P1 , P2 ), x := ⇡(x
P1 )
P2
D (P1 , P2 ), y := ⇡
and
1
(y
P2 )
P1
(4.35)
How do we analyze the security of this block cipher? Clearly for some ⇡’s this construction is insecure, for example when ⇡ is the identity function. For what ⇡ is EEM a secure block cipher? The only way we know to analyze security of EEM is by modeling ⇡ as a random permutation ⇧ on the set X (i.e., in the ideal cipher model using a fixed key). We show in Theorem 4.14 below that in the ideal cipher model, for all adversaries A: BCic adv[EEM , A]
2Qs Qic X 
(4.36)
where Qs is the number of queries A makes to EEM and Qic is the number of queries A makes to ⇧ and ⇧ 1 . Hence, the EvenMansour block cipher is secure (in the ideal cipher model) whenever X  is sufficiently large. Exercise 4.21 shows that the bound (4.36) is tight. The EvenMansour security theorem (Theorem 4.14) does not require the keys P1 and P2 to be independent. In fact, the bounds in (4.36) remain unchanged if we set P1 = P2 so that the key for EEM is a single element of X . However, we note that if one leaves out either of P1 or P2 , the construction is completely insecure (see Exercise 4.20). Iterated EvenMansour and AES. Looking back at our description of AES (Fig. 4.11) one observes that the EvenMansour cipher looks a lot like one round of AES where the round function ⇧AES plays the role of ⇡. Of course one round of AES is not a secure block cipher: the bound in (4.36) does not imply security because ⇧AES is not a random permutation. Suppose one replaces each occurrence of ⇧AES in Fig. 4.11 by a di↵erent permutation: one function for each round of AES. The resulting structure, called iterated EvenMansour, can be analyzed in the ideal cipher model and the resulting security bounds are better than those stated in (4.36). These results suggest a theoretical justification for the AES structure in the ideal cipher model. The EX construction and DESX. If we apply the EvenMansour construction to a fullfledged block cipher E = (E, D) defined over (K, X ), we obtain a new block cipher called EX = (EX, DX) where EX (k, P1 , P2 ), x := E(k, x
P1 )
P2 ,
DX (k, P1 , P2 ), y := D(k, y
P2 )
P1 . (4.37)
This new cipher EX has a key space K ⇥ X 2 which can be much larger than the key space for the underlying cipher E. Theorem 4.14 below shows that — in the ideal cipher model — this larger key space translates to better security: the maximum advantage against EX is much smaller than the maximum advantage against E, whenever X  is sufficiently large. Applying EX to the DES block cipher gives an efficient method to immunize DES against exhaustive search attacks. With P1 = P2 we obtain a block cipher called DESX whose key size is 56 + 64 = 120 bits: enough to resist exhaustive search. Theorem 4.14 shows that attacks in the 155
ideal cipher model on the resulting cipher are impractical. Since evaluating DESX requires only one call to DES, the DESX block cipher is three times faster than the TripleDES block cipher and this makes it seem as if DESX is the preferred way to strengthen DES. However, non blackbox attacks like di↵erential and linear cryptanalysis still apply to DESX where as they are ine↵ective against TripleDES. Consequently, DESX should not be used in practice.
4.7.4
Proof of the EvenMansour and EX theorems
We shall prove security of the EvenMansour block cipher (4.35) in the ideal permutation model and of the EX construction (4.37) in the ideal cipher model. We prove their security in a single theorem below. Taking a singlekey block cipher (i.e., K = 1) proves security of EvenMansour in the ideal permutation model. Taking a block cipher with a larger key space proves security of EX. Note that the pads P1 and P2 need not be independent and the theorem holds if we set P2 = P1 . Theorem 4.14. Let E = (E, D) be a block cipher defined over (K, X ). Let EX = (EX, DX) be the block cipher derived from E as in construction (4.37), where P1 and P2 are each uniformly distributed over a subset of X 0 of X . If we model E as an ideal cipher, and if A is an adversary in Attack Game 4.1 for EX that makes at most Qs standard queries (i.e., EXqueries) and Qic ideal cipher queries (i.e., ⇧ or ⇧ 1 queries), then we have BCic adv[A, EX]
2Qs Qic . KX 0 
(4.38)
2
To understand the security benefit of the EX construction consider the following: modeling E as an ideal cipher gives BCic adv[A, E] Qic /K for all A. Hence, Theorem 4.14 shows that, in the ideal cipher model, applying EX to E shrinks the maximum advantage by a factor of 2Qs /X 0 . The bounds in Theorem 4.14 are tight: there is an adversary A that achieves the advantage shown in (4.38); see Exercise 4.21. The advantage of this A is unchanged even when P1 and P2 are chosen independently. Therefore, we might as well always choose P2 = P1 . We also note that it is actually no harder to prove that EX is a strongly secure block cipher (see Section 4.1.3) in the ideal cipher model, with exactly the same security bounds as in Theorem 4.14. Proof idea. The basic idea is to show that the ideal cipher queries and the standard queries do not interact with each other, except with probability as bounded in (4.38). Indeed, to make the two types of queries interact with each other, the adversary has to make (k = k and a = x
P1 ) or (k = k and b = y
P2 )
for some input/output pair (x, y) corresponding to a standard query and some input/output triple (k , a , b ) corresponding to an ideal cipher query. Essentially, the adversary will have to simultaneously guess the random key k as well as one of the random pads P1 or P2 . Assuming there are no such interactions, we can e↵ectively realize all of the standard queries as ⇧(x P1 ) P2 using a random permutation ⇧ that is independent of the random permutations used to realize the ideal cipher queries. But ⇧0 (x) := ⇧(x P1 ) P2 is just a random permutation. Before giving a rigorous proof of Theorem 4.14, we present a technical lemma, called the Domain Separation Lemma, that will greatly simplify the proof, and is useful in analyzing other constructions. 156
To motivate the lemma, consider the following two experiments. In the one experiment, called the “split experiment”, an adversary has oracle access to two random permutations ⇧1 , ⇧2 on a set X . The adversary can make a series of queries, each of the form (µ, d, z ), where µ 2 {1, 2} specifies which of the two permutations to evaluate, d 2 {±1} specifies the direction to evaluate the permutation, and z 2 X the input to the permutation. On such a query, the challenger responds with z 0 := ⇧dµ (z ). Another experiment, called the “coalesced experiment”, is exactly the same as the split experiment, except that there is only a single permutation ⇧, and the challenger answers the query (µ, d, z ) with z 0 := ⇧d (z ), ignoring completely the index µ. The question is: under what condition can the adversary distinguish between these two experiments? Obviously, if the adversary can submit a query (1, +1, a ) and a query (2, +1, a ), then in the split experiment, the results will almost certainly be di↵erent, while in the coalesced experiment, they will surely be the same. Another type of attack is possible as well: the adversary could make a query (1, +1, a ) obtaining b , and then submit the query (2, 1, b ), obtaining a 0 . In the split experiment, a and a 0 will almost certainly be di↵erent, while in the coalesced experiment, they will surely be the same. Besides these two examples, one could get two more examples which reverse the direction of all the queries. The Domain Separation Lemma will basically say that unless the adversary makes queries of one of these four types, he cannot distinguish between these two experiments. Of course, the Domain Separation Lemma is only useful in contexts where the adversary is somehow constrained so that he cannot freely make queries of his choice. Indeed, we will only use it inside of the proof of a security theorem where the “adversary” in the Domain Separation Lemma comprises components of a challenger and an adversary in a more interesting attack game. In the more general statement of the lemma, we replace ⇧1 and ⇧2 by a family of permutations of permutations {⇧µ }µ2U , and we replace ⇧ by a family {⇧⌫ }⌫2V . We also introduce a function f : U ! V that specifies how several permutations in the split experiment are collapsed into one permutation in the coalesced experiment: for each ⌫ 2 V , all the permutations ⇧µ in the split experiment for which f (µ) = ⌫ are collapsed into the single permutation ⇧⌫ in the coalesced experiment. In the generalized version of the distinguishing game, if the adversary makes a query (µ, d, z ), then in the split experiment, the challenger responds with z 0 := ⇧dµ (z ), while in the coalesced experiment, the challenger responds with z 0 := ⇧df (µ) (z ). In the split experiment, we also keep track of the subset of the domains and ranges of the permutations that correspond to actual (d) queries made by the adversary in the split experiment. That is, we build up sets Domµ for each (+1) if and only if the adversary issues a query of the form µ 2 U and d 2 ±1, so that a 2 Domµ ( 1) if and only if the (µ, +1, a ) or a query of the form (µ, 1, b ) that yields a . Similarly, b 2 Domµ adversary issues a query of the form (µ, 1, b ) or a query of the form (µ, +1, a ) that yields b . We (+1) ( 1) call Domµ the sampled domain of ⇧µ and Domµ the sampled range of ⇧µ . Attack Game 4.5 (domain separation). Let U, V, X be finite, nonempty sets, and let f : U ! V be a function. For a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define: Experiment b: • For each µ 2 U , and each ⌫ 2 V the challenger sets ⇧µ R Perms[X ] and ⇧⌫ (d) Also, for each µ 2 U and d 2 {±1} the challenger sets Domµ ;. • The adversary submits a sequence of queries to the challenger. 157
R
Perms[X ]
For i = 1, 2, . . . , the ith query is (µi , di , z i ) 2 U ⇥ {±1} ⇥ X . If b = 0: the challenger sets z 0i
If b = 1: the challenger sets z 0i (d )
d
⇧fi(µi ) (z i ).
⇧dµii (z i ); the challenger also adds the value z i to the set ( di )
Domµii , and adds the value z 0i to the set Domµi
.
In either case, the challenger then sends z 0i to the adversary. • Finally, the adversary outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s domain separation distinguishing advantage as Pr[W0 ] Pr[W1 ]. We also define the domain separation failure event Z to be the event that in Experiment 1, at the end of the game we (d) (d) have Domµ \ Domµ0 6= ; for some d 2 {±1} and some pair of distinct indices µ, µ0 2 U with f (µ) = f (µ0 ). Finally, we define the domain separation failure probability to be Pr[Z]. 2 Experiment 1 is the above game is the split experiment and Experiment 0 is the coalesced experiment. Theorem 4.15 (Domain Separation Lemma). In Attack Game 4.5, an adversary’s domain separation distinguishing advantage is bounded by the domain separation failure probability. In the applying the Domain Separation Lemma, we will typically analyze some attack game in which permutations start out as coalesced, and then force them to be separated. We can bound the impact of this change on the outcome of the attack by analyzing the domain separation failure probability in the attack game with the split permutations. Before proving the Domain Separation Lemma, it is perhaps more instructive to see how it is used in the proof of Theorem 4.14. Proof of Theorem 4.14. Let A be an adversary as in the statement of the theorem. For b = 0, 1 let pb be the probability that A outputs 1 in Experiment b of the block cipher attack game in the ideal cipher model (Attack Game 4.1). So by definition we have BCic adv[A, EX] = p0
p1 .
(4.39)
We shall prove the theorem using a sequence of two games, applying the Domain Separation Lemma. Game 0. We begin by describing Game 0, which corresponds to Experiment 0 of the block cipher attack game in the ideal cipher model. Recall that in this model, we have a family of random permutations, and the encryption function is implemented in terms of this family. Also recall that in addition to standard queries that probe the function Ek (·), the adversary may also probe the random permutations. Initialize: for each k 2 K, set ⇧k k R K, choose P1 , P2
R
Perms[X ]
158
standard EXquery x: a x P1 1. 2. b ⇧k (a ) b P2 3. y 4. return y ideal cipher ⇧query k , a : 1. b ⇧k (a ) 2. return b ideal cipher ⇧ 1 query k , b : a ⇧k 1 (b ) 1. 2. return a Let W0 be the event that A outputs 1 at the end of Game 0. It should be clear from construction that (4.40) Pr[W0 ] = p0 . Game 1. In this game, we apply the Domain Separation Lemma. The basic idea is that we will declare “by fiat” that the random permutations used in processing the standard queries are independent of the random permutations used in processing ideal cipher queries. E↵ectively, each permutation ⇧k gets split into two independent permutations: ⇧std,k , which is used by the challenger in responding to standard EXqueries, and ⇧ic,k , which is used in responding to ideal cipher queries. In detail (changes from Game 0 are highlighted): Initialize: for each k 2 K, set ⇧std,k k R K, choose P1 , P2 standard EXquery x: a x P1 1. 2. b ⇧std,k (a ) // 3. y b P2 4. return y
R
Perms[X ] and ⇧ic,k
R
Perms[X ]
add a to sampled domain of ⇧std,k , add b to sampled range of ⇧std,k
ideal cipher ⇧query k , a : b ⇧ic,k (a ) // add a to sampled domain of ⇧ic,k , add b to sampled range of ⇧ic,k 1. 2. return b ideal cipher ⇧
1 query
⇧ic,1k (b )
1.
a
2.
return a
//
k , b: add a to sampled domain of ⇧ic,k , add b to sampled range of ⇧ic,k
Let W1 be the event that A outputs 1 at the end of Game 1. Let Z be the event that in Game 1 there exists k 2 K, such that the sampled domains of ⇧ic,k and ⇧std,k overlap or the sampled ranges 159
of ⇧ic,k and ⇧std,k overlap. The Domain Separation Lemma says that Pr[W0 ]
Pr[W1 ] Pr[Z].
(4.41)
In applying the Domain Separation Lemma, the “coalescing function” f maps from {std, ic} ⇥ K to K, sending the pair (·, k ) to k . Observe that the challenger only makes queries to ⇧k , where k is the secret key, and so such an overlap can occur only at k = k. Also observe that in Game 1, the random variables k, P1 , and P2 are completely independent of the adversary’s view. So the event Z occurs if and only if for some input/output triple (k , a , b ) triple arising from a ⇧ or ⇧ 1 query, and for some input/output pair (x, y) arising from an EXquery, we have (k = k and a = x
P1 ) or (k = k and b = y
P2 ).
(4.42)
Using the union bound, we can therefore bound Pr[Z] as a sum of probabilities of 2Qs Qic events, each of the form k = k and a = x P1 , or of the form k = k and b = y P2 . By independence, since k is uniformly distributed over a set of size K, and each of P1 and P2 is uniformly distributed over a set of size X 0 , each such event occurs with probability at most 1/(KX 0 ). It follows that Pr[Z]
2Qs Qic . KX 0 
(4.43)
Finally, observe that Game 1 is equivalent to Experiment 1 of the block cipher attack game in the ideal cipher model: the EXqueries present to the adversary the random permutation ⇧0 (x) := ⇧std,k (x P1 ) P2 and this permutation is independent of the random permutations used in the ⇧ and ⇧ 1 queries. Thus, (4.44) Pr[W1 ] = p1 . The bound (4.38) now follows from (4.39), (4.40), (4.41), (4.43), and (4.44). This completes the proof of the theorem. 2 Finally, we turn to the proof of the Domain Separation Lemma, which is a simple (if tedious) application of the Di↵erence Lemma and the “forgetful gnome” technique. Proof of Theorem 4.15. We define a sequence of games. Game 0. This game will be equivalent to the coalesced experiment in Attack Game 4.5, but designed in a way that will facilitate the analysis. In this game, the challenger maintains various sets ⇧ of pairs (a , b ). Each set ⇧ represents a function that can be extended to a permutation on X that sends a to b for every (a , b ) in ⇧. We call such a set ⇧ a partial permutation on X . Define Domain(⇧) = {a 2 X : (a , b ) 2 ⇧ for some b 2 X } , Range(⇧) = {b 2 X : (a , b ) 2 ⇧ for some a 2 X } .
Also, for a 2 Domain(⇧), define ⇧(a ) to be the unique b such that (a , b ) 2 ⇧. Likewise, for b 2 Range(⇧), define ⇧ 1 (b ) to be the unique a such that (a , b ) 2 ⇧. Here is the logic of the challenger in Game 0: Initialize: for each ⌫ 2 V , initialize the partial permutation ⇧⌫ 160
;
Process query (µ, +1, a ): 1. if a 2 Domain(⇧f (µ) ) then b 2. b R X \ Range(⇧f (µ) ) 3. add (a , b ) to ⇧f (µ) 4. return b Process query (µ, 1, b ): 1. if b 2 Range(⇧f (µ) ) then a 2. a R X \ Domain(⇧f (µ) ) 3. add (a , b ) to ⇧µ 4. return a
⇧f (µ) (a ), return b
1
⇧f (µ) (b ), return a
This game is clearly equivalent to the coalesced experiment in Attack Game 4.5. Let W0 be the event that the adversary outputs 1 in this game. Game 1. Now we modify this game to get an equivalent game, but it will facilitate the application of the Di↵erence Lemma in moving to the next game. For µ, µ0 2 U , let us write µ ⇠ µ0 if f (µ) = f (µ0 ). The is an equivalence relation on U , and we write [µ] for the equivalence class containing µ. Here is the logic of the challenger in Game 1: Initialize: for each µ 2 U , initialize the partial permutation ⇧µ
;
Process query (µ, +1, a ): ⇧µ (a ), return b 1a. if a 2 Domain(⇧µ ) then b ⇤ 1b. if a 2 Domain(⇧ 0 ) for some µ0 2 [µ] then b ⇧µ0 (a ), return b µ R 2a. b XS\ Range(⇧µ ) S R ⇤ 2b. if b 2 X \ µ0 2[µ] Range(⇧µ0 ) µ0 2[µ] Range(⇧µ0 ) then b 3. add (a , b ) to ⇧µ 4. return b Process query (µ, 1, b ): ⇧µ 1 (b ), return a 1a. if b 2 Range(⇧µ ) then a ⇤ 1b. if b 2 Range(⇧ 0 ) for some µ0 2 [µ] then a ⇧µ01 (b ), return a µ 2a. a R XS\ Domain(⇧µ ) S R ⇤ 2b. if a 2 X \ µ0 2[µ] Domain(⇧µ0 ) µ0 2[µ] Domain(⇧µ0 ) then a 3. add (a , b ) to ⇧µ 4. return a Let W1 be the event that the adversary outputs 1 in this game. It is not hard to see that the challenger’s behavior in this game is equivalent to that in Game 0, and so Pr[W0 ] = Pr[W1 ]. The idea is that for every ⌫ 2 f (U ) ✓ V , the partial permutation ⇧⌫ in Game 0 is partitioned into a family of disjoint partial permutations {⇧µ }µ2f 1 (⌫) , so that ⇧⌫ =
[
µ2f
⇧µ ,
1 (⌫)
161
and Domain(⇧µ ) \ Domain(⇧µ0 ) = ; and for all µ, µ0 2 f 1 (⌫) with µ 6= µ0 .
Range(⇧µ ) \ Range(⇧µ0 ) = ;
(4.45)
Game 2. Now we simply delete the lines marked with a “⇤ ” in Game 1. Let W2 be the event that the adversary outputs 1 in this game. It is clear that this game is equivalent to the split experiment in Attack Game 4.5, and so Pr[W2 ] Pr[W1 ] is equal to the adversary’s advantage in Attack Game 4.5. We want to use the Di↵erence Lemma to bound Pr[W2 ] Pr[W1 ]. To make this entirely rigorous, one models both games as operating on the same underlying probability space: we define a collection of random variables representing the coins of the adversary, as well as the various random samples from di↵erent subsets of X made by the challenger. These random variables completely describe both Games 1 and 2: the only di↵erence between the two games are the deterministic computation rules that determine the outcomes. Define Z be to be the event that at the end of Game 2, the condition (4.45) does not hold. One can verify that Games 1 and 2 proceed identically unless Z holds, so by the Di↵erence Lemma, we have Pr[W2 ] Pr[W1 ] Pr[Z]. Moreover, it is clear that Pr[Z] is precisely the failure probability in Attack Game 4.5. 2
4.8
Fun application: comparing information without revealing it
In this section we describe an important application for PRFs called subkey derivation. Alice and Bob have a shared key k for a PRF. They wish to generate a sequence of shared keys k1 , k2 , . . . so that key number i can be computed without having to compute all earlier keys. Naturally, they set ki := F (k, i) where F is a secure PRF whose input space is {1, 2, . . . , B} for some bound B. The generated sequence of keys is indistinguishable from random keys. As a fun application of this, consider the following problem: Alice is on vacation at the Squaw valley ski resort and wants to know if her friend Bob is also there. If he is they could ski together. Alice could call Bob and ask him if he is on the slopes, but this would reveal to Bob where she is and Alice would rather not do that. Similarly, Bob values his privacy and does not want to tell Alice where he is, unless Alice happens to be close by. Abstractly, this problem can be phrased as follows: Alice has a number a 2 Zp and Bob has a number b 2 Zp for some prime p. These numbers indicate their approximate positions on earth. Think of dividing the surface of the earth into p squares and the numbers a and b indicate what square Alice and Bob are currently at. If Bob is at the resort then a = b, otherwise a 6= b. Alice wants to learn if a = b; however, if a 6= b then Alice should learn nothing else about b. Bob should learn nothing at all about a. In a later chapter we will see how to solve this exact problem. Here, we make the problem easier by allowing Alice and Bob to interact with a server, Sam, that will help Alice learn if a = b, but will itself learn nothing at all. The only assumption about Sam is that it does not collude with Alice or Bob, that is, it does not reveal private data that Alice or Bob send to it. Clearly, Alice and Bob could send a and b to Sam and he will tell Alice if a = b, but then Sam would learn both a and b. Our goal is that Sam learns nothing, not even if a = b. To describe the basic protocol, suppose Alice and Bob have a shared secret key (k0 , k1 ) 2 Z2p . Moreover, Alice and Bob each have a private channel to Sam. The protocol for comparing a and b is shown in Fig. 4.17. It begins with Bob choosing a random r in Zp and sending (r, xb ) to Sam. 162
Alice input: a
Server Sam xa
?
x
a + k0 r xa
xb
Bob input: b r,
xb
r(b + k0 ) + k1
!
r
R
Zp
x + k1 = 0 Figure 4.17: Comparing a and b without revealing them Bob can do this whenever he wants, even before Alice initiates the protocol. When Alice wants to r xa xb and sends x back to Alice. Now, test equality, she sends xa to Sam. Sam computes x observe that x + k1 = r(a b) so that x + k1 = 0 when a = b and x + k1 is very likely to be nonzero otherwise (assuming p is sufficiently large so that r 6= 0 with high probability). This lets Alice learn if a = b. What is revealed by this protocol? Clearly Bob learns nothing. Alice learns r(a b), but if a 6= b this quantity is uniformly distributed in Zp . Therefore, when a 6= b Alice just obtains a uniform element in Zp and this reveals nothing beyond the fact that a 6= b. Sam sees r, xa , xb , but all three values are independent of a and b: xa and xb are onetime pad encryptions under keys k0 and k1 , respectively. Therefore, Sam learns nothing. Notice that the only privacy assumption about Sam is that it does not reveal (r, xb ) to Alice or xa to Bob. The trouble, much like with the onetime pad, is that the shared key (k0 , k1 ) can only be used for a single equality test, otherwise the protocol becomes insecure. If (k0 , k1 ) is used to test if a = b and later the same key (k0 , k1 ) is used to test if a0 = b0 then Alice and Sam learn information they are not supposed to. For example, Sam learns a a0 . Moreover, Alice can deduce (a b)/(a0 b0 ) which reveals information about b and b0 (e.g., if a = a0 = 0 then Alice learns the ratio of b and b0 ). Subkey derivation. What if Alice wants to repeatedly test proximity to Bob? The solution is to generate a new independent key (k0 , k1 ) for each invocation of the protocol. We do so by deriving instancespecific subkeys using a secure PRF. Let F be a secure PRF defined over (K, {1, . . . , B}, Z2p ) and suppose that Alice and Bob share a long term key k 2 K. Bob maintains a counter cnt b that is initially set to 0. Every time Bob sends his encrypted location (r, xb ) to Sam he increments cnt b and derives subkeys (k0 , k1 ) from the longterm key k as: F (k, cnt b ). (4.46) (k0 , k1 ) He sends (r, xb , cnt b ) to Sam. Bob can do this whenever he wants, say every few minutes, or every time he moves to a new location. Whenever Alice wants to test proximity to Bob she first asks Sam to send her the value of the counter in the latest message from Bob. She makes sure the counter value is larger than the previous value Sam sent her (to prevent a mischievous Sam or Bob from tricking Alice into reusing an old counter value). Alice then computes (k0 , k1 ) herself using (4.46) and carries out the protocol with Sam in Fig. 4.17 using these keys.
163
Because F is a secure PRF, the sequence of derived subkeys is indistinguishable from random independently sampled keys. This ensures that the repeated protocol reveals nothing about the tested values beyond equality. By using a PRF, Alice is able to quickly compute (k0 , k1 ) for the latest value of cnt b .
4.9
Notes
Citations to the literature to be added.
4.10
Exercises
4.1 (Exercising the definition of a secure PRF). Let F be a secure PRF defined over (K, X , Y), where K = X = Y = {0, 1}n . (a) Show that F1 (k, x) = F (k, x) k 0 is not a secure PRF. (b) Prove that F2 k, (x, y) := F (k, x) (c) Prove that F3 (k, x) := F (k, x)
F (k, y) is insecure.
x is a secure PRF.
(d) Prove that F4 (k1 , k2 ), x := F (k1 , x) (e) Show that F5 (k, x) := F (k, x) k F (k, x
F (k2 , x) is a secure PRF. 1n ) is insecure.
(f) Prove that F6 (k, x) := F (F (k, 0n ), x) is a secure PRF. (g) Show that F7 (k, x) := F (F (k, 0n ), x) k F (k, x) is insecure. (h) Show that F8 (k, x) := F (k, x) k F k, F (k, x) is insecure. 4.2 (Weak PRFs). Let F be a PRF defined over (K, X , Y) where Y := {0, 1}n and X  is superpoly. Define F2 k, (x, y) := F (k, x) F (k, y). We showed in Exercise 4.1 part (b) that F2 is not a secure PRF. (a) Show that F2 is a weakly secure PRF (as in Definition 4.3), assuming F is weakly secure. In particular, for any Qquery weak PRF adversary A attacking F2 (i.e., an adversary that only queries the function at random points in X ) there is a weak PRF adversary B attacking F , where B is an elementary wrapper around A, such that wPRFadv[A, F2 ] wPRFadv[B, F ] + (Q/X )4 . (b) Suppose F is a secure PRF. Show that F2 is weakly secure even if we modify the weak PRF attack game and allow the adversary A to query F2 at one chosen point in addition to the Q random points. A PRF that is secure in this sense is sufficient for a popular data integrity mechanism discussed in Section 7.4. (c) Show that F2 is no longer secure if we modify the weak PRF attack game and allow the adversary A to query F2 at two chosen points in addition to the Q random points. 164
4.3 (Format preserving encryption). Suppose we are given a block cipher (E, D) operating on domain X . We want a block cipher (E 0 , D0 ) that operates on a smaller domain X 0 ✓ X . Define (E 0 , D0 ) as follows: E 0 (k, x) :=
y E(k, x) while y 62 X 0 do: y output y
E(k, y)
D0 (k, y) is defined analogously, applying D(k, ·) until the result falls in X 0 . Clearly (E 0 , D0 ) are defined on domain X 0 .
(a) With t := X /X 0 , how many evaluations of E are needed in expectation to evaluate E 0 (k, x) as a function of t? You answer shows that when t is small (e.g., t 2) evaluating E 0 (k, x) can be done efficiently.
(b) Show that if (E, D) is a secure block cipher with domain X then (E 0 , D0 ) is a secure block cipher with domain X 0 . Try proving security by induction on X  X 0 . Discussion: This exercise is used in the context of encrypted 16digit credit card numbers where the ciphertext also must be a 16digit number. This type of encryption, called format preserving encryption, amounts to constructing a block cipher whose domain size is exactly 1016 . This exercise shows that it suffices to construct a block cipher (E, D) with domain size 254 which is the smallest power of 2 larger than 1016 . The procedure in the exercise can then be used to shrink the domain to size 1016 . 4.4 (Truncating PRFs). Let F be a PRF whose range is Y = {0, 1}n . For some ` < n consider the PRF F 0 with a range Y 0 = {0, 1}` defined as: F 0 (k, x) = x[0 . . . ` 1]. That is, we truncate the output of F (k, x) to the first ` bits. Show that if F is a secure PRF then so is F 0 . 4.5 (Twokey TripleDES). Consider the following variant of the 3E construction that uses only two keys: for a block cipher (E, D) with key space K define 3E 0 as E((k1 , k2 ), m) := E(k1 , E(k2 , E(k1 , m))). Show that this block cipher can be defeated by a meet in the middle attack using O(K) evaluation of E and D and using O(K) encryption queries to the block cipher challenger. Further attacks on this method are discussed in [74, 68]. 4.6 (adaptive vs nonadaptive security). This exercise develops an argument that shows that a PRF may be secure against every adversary that makes its queries nonadaptively, (i.e., all at once) but is insecure against adaptive adversaries (i.e., the kind allowed in Attack Game 4.2). To be a bit more precise, we define the nonadaptive version of Attack Game 4.2 as follows. The adversary submits all at once the query (x1 , . . . , xQ ) to the challenger, who responds with (y1 , . . . , yQ ), where y := f (xi ). The rest of the attack game is the same: in Experiment 0, k R K and f R F (k, ·), while in Experiment 1, f R Funs[X , Y]. Security against nonadaptive adversaries means that all efficient adversaries have only negligible advantage; advantage is defined as usual: Pr[W0 ] Pr[W1 ], where Wb is the event that the adversary outputs 1 in Experiment b. Suppose F is a secure PRF defined over (K, X , X ), where N := X  is superpoly. We proceed to “sabotage” F , constructing a new PRF F˜ as follows. Let x0 be some fixed element of X . For x = F (k, x0 ) define F˜ (k, x) := x0 , and for all other x define F˜ (k, x) := F (k, x). 165
(a) Show that F˜ is not a secure PRF against adaptive adversaries. (b) Show that F˜ is a secure PRF against nonadaptive adversaries. (c) Show that a similar construction is possible for block ciphers: given a secure block cipher (E, D) defined over (K, X ) where X  is superpoly, construct a new, “sabotaged” block ˜ D) ˜ that is secure against nonadaptive adversaries, but insecure against adaptive cipher (E, adversaries. 4.7 (PRF security definition). This exercise develops an alternative characterization of PRF security for a PRF F defined over (K, X , Y). As usual, we need to define an attack game between an adversary A and a challenger. Initially, the challenger generates b
R
{0, 1}, k
R
K, y1
R
Y
Then A makes a series of queries to the challenger. There are two types of queries: Encryption: In an function query, A submits an x 2 X to the challenger, who responds with y F (k, x). The adversary may make any (polybounded) number of function queries. Test: In a test query, A submits an x 2 X to the challenger, who computes y0 F (k, x) and responds with yb . The adversary is allowed to make only a single test query (with any number of function queries before and after the test query). At the end of the game, A outputs a bit ˆb 2 {0, 1}. As usual, we define A’s advantage in the above attack game to be Pr[ˆb = b] 1/2. We say that F is AltPRF secure if this advantage is negligible for all efficient adversaries. Show that F is a secure PRF if and only if F is AltPRF secure.
Discussion: This characterization shows that the value of a secure PRF at a point x0 in X looks like a random element of Y, even after seeing the value of the PRF at many other points of X . 4.8 (Key malleable PRFs). Let F be a PRF defined over ({0, 1}n , {0, 1}n , Y). (a) We say that F is XORmalleable if F (k, x
c) = F (k, x)
(b) We say that F is key XORmalleable if F (k
c for all k, x, c in {0, 1}n .
c, x) = F (k, x)
c for all k, x, c in {0, 1}n .
Clearly an XORmalleable PRF cannot be secure: malleability lets an attacker distinguish the PRF from a random function. Show that the same holds for a key XORmalleable PRF. Remark: In contrast, we note that there are secure PRFs where F (k1 k2 , x) = F (k1 , x) F (k2 , x). See Exercise 11.1 for an example, where the xor on the left is replaced by addition, and the xor on the right is replaced by multiplication. 4.9 (Strongly secure block ciphers). In Section 4.1.3 we sketched out the notion of a strongly secure block cipher. (a) Write out the complete definition of a strongly secure block cipher as a game between a challenger and an adversary. (b) Consider the following cipher E 0 = (E 0 , D0 ) built from a block cipher (E, D) defined over (K, {0, 1}n ): E 0 (k, m) := D(k, t
E(k, m) )
and
166
D0 (k, c) := E(k, t
D(k, m) )
where t 2 {0, 1}n is a fixed constant. For what values of t is this cipher E 0 semantically secure? Prove semantic security assuming the underlying block cipher is strongly secure. 4.10 (Meetinthemiddle attacks). Let us study the security of the 4E construction where a block cipher (E, D) is iterated four times using four di↵erent keys: E4 ( (k1 , k2 , k3 , k4 ), m) = E k4 , E(k3 , E(k2 , E(k1 , m))) where (E, D) is a block cipher with key space K. (a) Show that there is a meet in the middle attack on 4E that recovers the secret key in time K2 and memory space K2 .
(b) Show that there is a meet in the middle attack on 4E that recovers the secret key in time K2 , but only uses memory space K. If you get stuck see [32]. 4.11 (Tweakable block ciphers). A tweakable block cipher is a block cipher whose encryption and decryption algorithm take an additional input t, called a “tweak”, which is drawn from a “tweak space” T . As usual, keys come from a key space K, and data blocks from a data block space X . The encryption and decryption functions operate as follows: for k 2 K, x 2 X , t 2 T , we have y = E(k, x, t) 2 X and x = D(k, y, t). So for each k 2 K and t 2 T , E(k, ·, t) defines a permutation on X and D(k, ·, t) defines the inverse permutation. Unlike keys, tweaks are typically publicly known, and may even be adversarially chosen. Security is defined by a game with two experiments. In both experiments, the challenger defines a family of permutations {⇧t }t2T , where each ⇧t is a permutation on X . In Experiment 0, the challenger sets k R K, and ⇧t := E(k, ·, t) for all t 2 T . In Experiment 1, the challenger sets
⇧t
R
Perms[X ] for all t 2 T .
Both experiments then proceed identically. The adversary issues a series of queries. Each query is one of two types: forward query: the adversary sends (x, t) 2 X ⇥ T , and the challenger responds with y := ⇧t (x); inverse queries: the adversary sends (y, t) 2 X ⇥ T , and the challenger responds with x := ⇧t 1 (y). At the end of the game, the adversary outputs a bit. If pb is the probability that the adversary outputs 1 in Experiment b, the adversary’s advantage is defined to be p0 p1 . We say that (E, D) is a secure tweakable block cipher if every efficient adversary has negligible advantage. This definition of security generalizes the notion of a strongly secure block cipher (see Section 4.1.3 and Exercise 4.9). In applications of tweakable block ciphers, this strong security notion is more appropriate (e.g., see Exercise 9.17). ˜ m, t) := E(E(k, t), m) where (E, D) is a strongly (a) Prove security of the construction E(k, secure block cipher defined over (K, K). (b) Show that there is an attack on the construction from part (a) that achieves advantage 1/2 p and which makes Q ⇡ K queries. p p Hint: In addition to the ⇡ K queries, your adversary should make an additional ⇡ K “o✏ine” evaluations of the cipher (E, D). 167
(c) Prove security of the construction E 0 (k0 , k1 ), m, t := p
F (k0 , t); output p
E(k1 , m
p) ,
where (E, D) is a strongly secure block cipher and F is a secure PRF. In Exercise 7.10 we will see a more efficient variant of this construction. Hint: Use the assumption that (E, D) is a strongly secure block cipher to replace E(k1 , ·) in e then, use the Domain Separation Lemma the challenger by a truly random permutation ⇧; e e t }t2T , and (see Theorem 4.15) to replace ⇧ by a family of independent permutations {⇧ analyze the corresponding domain separation failure probability.
Discussion: Tweakable block ciphers are used in disk sector encryption where encryption must not expand the data: the ciphertext size is required to have the same size as the input. The sector number is used as the tweak to ensure that even if two sectors contain the same data, the resulting encrypted sectors are di↵erent. The construction in part (c) is usually more efficient than that in part (a), as the latter uses a di↵erent block cipher key with every evaluation, which can incur extra costs. See further discussion in Exercise 7.10.
4.12 (PRF combiners). We want to build a PRF F using two PRFs F1 and F2 , so that if at some future time one of F1 or F2 is broken (but not both) then F is still secure. Put another way, we want to construct F from F1 and F2 such that F is secure if either F1 or F2 is secure. Suppose F1 and F2 both have output spaces {0, 1}n , and both have a common input space. Define F ( (k1 , k2 ), x) := F1 (k1 , x)
F2 (k2 , x).
Show that F is secure if either F1 or F2 is secure. 4.13 (Block cipher combiners). Continuing with Exercise 4.12, we want to build a block cipher E = (E, D) from two block ciphers E1 = (E1 , D1 ) and E2 = (E2 , D2 ) so that if at some future time one of E1 or E2 is broken (but not both) then E is still secure. Suppose both E1 and E2 are defined over (K, X ). Define E as: E( (k1 , k2 ), x) := E1 k1 , E2 (k2 , x)
and
D( (k1 , k2 ), y) := D2 k2 , D1 (k1 , y) .
(a) Show that E is secure if either E1 or E2 is secure. (b) Show that this is not a secure combiner for PRFs. That is, F ( (k1 , k2 ), x) := F1 k1 , F2 (k2 , x) need not be a secure PRF even if one of F1 or F2 is. 4.14 (Key leakage). Let F be a secure PRF defined over (K, X , Y), where K = X = Y = {0, 1}n .
(a) Let K1 = {0, 1}n+1 . Construct a new PRF F1 , defined over (K1 , X , Y), with the following property: the PRF F1 is secure; however, if the adversary learns the last bit of the key then the PRF is no longer secure. This shows that leaking even a single bit of the secret key can completely destroy the PRF security property. Hint: Let k1 = k k b where k 2 {0, 1}n and b 2 {0, 1}. Set F1 (k1 , x) to be the same as F (k, x) for all x 6= 0n . Define F1 (k1 , 0n ) so that F1 is a secure PRF, but becomes easily distinguishable from a random function if the last bit of the secret key k1 is known to the adversary. 168
(b) Construct a new PRF F2 , defined over (K ⇥ K, X , Y), that remains secure if the attacker learns any single bit of the key. Your function F2 may only call F once. 4.15 (Variants of LubyRacko↵ ). Let F be a secure PRF defined over (K, X , X ). (a) Show that tworound LubyRacko↵ is not a secure block cipher. (b) Show that threeround LubyRacko↵ is not a strongly secure block cipher. 4.16 (Insecure tree construction). In the tree construction for building a PRF from a PRG (Section 4.6), the secret key is used at the root of the tree and the input is used to trace a path through the tree. Show that a construction that does the opposite is not a secure PRF. That is, using the input as the root and using the key to trace through the tree is not a secure PRF. 4.17 (Truncated tree construction). Suppose we cut o↵ the tree construction from Section 4.6 after only three levels of the tree, so that there are only eight leaves, as in Fig. 4.15. Give a direct proof, using a sequence of seven hybrids, that outputting the values at all eight leaves gives a secure PRG defined over (S, S 8 ), assuming the underlying PRG is secure. 4.18 (Augmented tree construction). Suppose we are given a PRG G defined over (K ⇥ S, S 2 ). Write G(k, s) = (G0 (k, s), G1 (k, s)). Let us define the PRF G⇤ with key space Kn ⇥ S and input space {0, 1}n as follows: G⇤ (k0 , . . . , kn 1 , s), x 2 {0, 1}n := t s for i 0 to n 1 do b x[i] t Gb (ki , t) output t. (a) Given an example secure PRG G for which G⇤ is insecure as a PRF. (b) Show that G⇤ is a secure PRF if for every polybounded Q the following PRG is secure: G0 (k, s0 , . . . , sQ
1)
:= (G(k, s0 ), . . . , G(k, sQ
1 ))
.
4.19 (Synthesizers and parallel PRFs). For a secure PRG G defined over (S, R) we showed that Gn (s1 , . . . , sn ) := G(s1 ), . . . , G(sn ) is a secure PRG over (S n , Rn ). The proof requires that the components s1 , . . . , sn of the seed be chosen uniformly and independently over S n . A secure synthesizer is a PRG for which this holds even if s1 , . . . , sn are not independent of one another. Specifically, a synthesizer is an efficient function S : X 2 ! X . The synthesizer is said to be nway secure if 2 S n (x1 , y1 , . . . , xn , yn ) := S(xi , yj ) i,j=1,...,n 2 X (n ) 2
is a secure PRG defined over (X 2n , X (n ) ). Here S is being evaluated at n2 inputs that are not independent of one another and yet S n is a secure PRG. (a) Not every secure PRG is a secure synthesizer. Let G be a secure PRG over (S, R). Show that S(x, y) := G(x), y is a secure PRG defined over (S 2 , R ⇥ S), but is an insecure 2way synthesizer. 169
¯ 00101011) F (k, S S
S
S
S
S
S
k00
k10
k21
k30
k41
k50
k61
k71
k00
k10
k20
k30
k40
k50
k60
k70
k01
k11
k21
k31
k41
k51
k61
k71
key k¯ 2 X 16
Figure 4.18: A PRF built from a synthesizer S. The PRF input in {0, 1}n is used to select n components from the key k¯ 2 X 2n . The selected components, shown as shaded squares, are used as shown in the figure. (b) A secure synthesizer lets us build a large domain PRF that can be evaluated quickly on a parallel computer. Show that if S : X 2 ! X is a Qway secure synthesizer, for polybounded Q, then the PRF in Fig. 4.18 is a secure PRF defined over (X 2n , {0, 1}n , X ). For simplicity, assume that n is a power of 2. Observe that the PRF can be evaluated in only log2 n steps on a parallel computer. 4.20 (Insecure variants of EvenMansour). In Section 4.7.3 we discussed the EvenMansour block cipher (E, D) built from a permutation ⇡ : X ! X where X = {0, 1}n . Recall that E (P0 , P1 ), m := ⇡(m P0 ) P1 . (a) Show that E1 (P0 , m) := ⇡(m
P0 ) is not a secure block cipher.
(b) Show that E2 (P1 , m) := ⇡(m)
P1 is not a secure block cipher.
4.21 (Birthday attack on EvenMansour). Let’s show that the bounds in the EvenMansour security theorem (Theorem 4.14) are tight. For X := {0, 1}n , recall that the EvenMansour block cipher (E, D), built from a permutation ⇡ : X ! X , is defined as: E (k0 , k1 ), m := ⇡(m k0 ) k1 . We show how to break this block cipher in time approximately 2n/2 . (a) Show that for all a, m,
2 X and k¯ := (k0 , k1 ) 2 X 2 , whenever a = m ¯ m E k,
¯ m E k,
= ⇡(a)
⇡(a
k0 , we have
)
(b) Use part (a) to construct an adversary A that wins the block cipher security game against (E, D) with advantage close to 1, in the ideal cipher model. With q := 2n/2 and some nonzero 170
2 X , the adversary A queries the cipher at 2q random points mi , mi 2 X , for i = 1, . . . , q. the permutation ⇡ at 2q random points ai , ai
2 X and queries
4.22 (A variant of the EvenMansour cipher). Let M := {0, 1}m , K := {0, 1}n , and X := {0, 1}n+m . Consider the following cipher (E, D) defined over (K, M, X ) built from a permutation ⇡ : X ! X: (4.47) E(k, x) := (k k 0m ) ⇡(k k x) D(k, c) is defined analogously. Show that if we model ⇡ as an ideal permutation ⇧, then for every block cipher adversary A attacking (E, D) we have BCic adv[A, E]
2Qic . K
Here Qic is the number of queries A makes to ⇧ and ⇧
(4.48)
1 oracles.
4.23 (Analysis of Salsa and ChaCha). In this exercise we analyze the Salsa and ChaCha stream ciphers from Section 3.6 in the ideal permutation model. Let ⇡ : X ! X be a permutation, where X = {0, 1}n+m . Let K := {0, 1}n and define the PRF F , which is defined over (K, {0, 1}m , X ), as F (k, x) := (k k x) ⇡(k k x) . (4.49) This PRF is an abstraction of the PRF underlying the Salsa and ChaCha stream ciphers. Use Exercise 4.22 to show that if we model ⇡ as an ideal permutation ⇧, then for every PRF adversary A attacking F we have Q2 2Qic + F (4.50) PRFic adv[A, F ] K 2X 
where QF is the number of queries that A makes to an F (k, ·) oracle and Qic is the number of
queries A makes to ⇧ and ⇧ “negligible.”
1 oracles.
In Salsa and ChaCha, QF is at most X 1/4 so that
Q2F 2X 
is
Discussion: The specific permutation ⇡ used in the Salsa and ChaCha stream ciphers is not quite an ideal permutation. For example, ⇡(0n+m ) = 0n+m . Hence, your analysis applies to the general framework, but not specifically to Salsa and ChaCha. 4.24 (Alternative proof of Theorem 4.6). Let X and Y be random variables as defined in Exercise 3.13. Consider an adversary A in Attack Game 4.3 that makes at most Q queries to its challenger. Show that PFadv[A, X ] [X, Y] Q2 /2N . 4.25 (A onesided switching lemma). Following up on the previous exercise, one can use part (b) of Exercise 3.13 to get a “one sided” version of Theorem 4.6, which can be useful in some settings. Consider an adversary A in Attack Game 4.3 that makes at most Q queries to its challenger. Let W0 and W1 be as defined in that game: W0 is the event that A outputs 1 when probing a random permutation, and W1 is the event that A outputs 1 when probing a random function. Assume Q2 < N . Show that Pr[W0 ] ⇢[X, Y] · Pr[W1 ] 2 Pr[W1 ]. 4.26 (Parallel composition of PRFs). Just as we can compose PRGs in parallel, while maintaining security (see Section 3.4.1), we can also compose PRFs in parallel, while maintaining security. 171
Suppose we have a PRF F , defined over (K, X , Y). We want to model the situation where an adversary is given n black boxes (where n 1 is polybounded): the boxes either contain F (k1 , ·), . . . , F (kn , ·), where the ki are random (and independent) keys, or they contain f1 , . . . , fn , where the fi are random elements of Funs[X , Y], and the adversary should not be able to tell the di↵erence. A convenient way to model this situation is to consider the nwise parallel composition of F , which is a PRF F 0 whose key space is Kn , whose input space is {1, . . . , n} ⇥ X , and whose output space is Y. Given a key k 0 = (k1 , . . . , kn ), and an input x0 = (s, x), with s 2 {1, . . . , n} and x 2 X , we define F 0 (k 0 , x0 ) := F (ks , x). Show that if F is a secure PRF, then so is F 0 . In particular, show that for every PRF adversary A, then exist a PRF adversary B, where B is an elementary wrapper around A, such that PRFadv[A, F 0 ] = n · PRFadv[B, F ]. 4.27 (Universal attacker on PRFs). Let F be a PRF defined over (K, X , Y) where K < X . Let Q < K. Show that there is a PRF adversary A that runs in time proportional to Q, makes one query to the PRF challenger, and has advantage PRFadv[A, F ]
Q K
Q . X 
4.28 (Distributed PRFs). Let F be a secure PRF defined over (K, X , Y) where Y := {0, 1}n . In Exercise 4.1 part (d) we showed that if F is secure then so is F 0 (k1 , k2 ), x) := F (k1 , x)
F (k2 , x).
This F 0 has a useful property: the PRF key (k1 , k2 ) can be split into two shares, k1 and k2 . If Alice is given one share and Bob the other share, then both Alice and Bob are needed to evaluate the PRF, and neither can evaluate the PRF on its own. Moreover, the PRF can be evaluated distributively, that is, without reconstituting the key (k1 , k2 ): to evaluate the PRF at a point x0 , Alice simply sends F (k1 , x0 ) to Bob. (a) To show that Alice cannot evaluate F 0 by herself, show that F 0 is a secure PRF even if the adversary is given k1 . Argue that the same holds for k2 . (b) Construct a PRF where the key can be split into three shares s1 , s2 , s3 so that any two shares can be used evaluate the PRF distributively, but no single share is sufficient to evaluate the PRF on its own. Hint: Consider the PRF F 00 (k1 , k2 , k3 ), x) := F (k1 , x) F (k2 , x) F (k3 , x) and show how to construct the shares s1 , s2 , s3 from the keys k1 , k2 , k3 . Make sure to prove that the F 00 is a secure PRF when the adversary is given a single share, namely si for some i 2 {1, 2, 3}. (c) Generalize the construction from part (b) to construct a PRF F 000 supporting threeoutoffive sharing of the key: any three shares can be used to evaluate the PRF distributively, but no two shares can. Hint: The key space for F 000 is K10 .
172
Chapter 5
Chosen Plaintext Attack This chapter focuses on the problem of securely encrypting several messages in the presence of an adversary who eavesdrops, and who may even may influence the choice of some messages in order to glean information about other messages. This leads us to the notion of semantic security against a chosen plaintext attack.
5.1
Introduction
In Chapter 2, we focused on the problem of encrypting a single message. Now we consider the problem of encrypting several messages. To make things more concrete, suppose Alice wants to use a cipher to encrypt her files on some file server, while keeping her secret keys for the cipher stored securely on her USB memory stick. One possible approach is for Alice to encrypt each individual file using a di↵erent key. This entails that for each file, she stores an encryption of that file on the file server, as well as a corresponding secret key on her memory stick. As we will explore in detail in Section 5.2, this approach will provide Alice with reasonable security, provided she uses a semantically secure cipher. Now, although a file may be several megabytes long, a key for any practical cipher is just a few bytes long. However, if Alice has many thousands of files to encrypt, she must store many thousands of keys on her memory stick, which may not have sufficient storage for all these keys. As we see, the above approach, while secure, is not very space efficient, as it requires one key per file. Faced with this problem, Alice may simply decide to encrypt all her files with the same key. While more efficient, this approach may be insecure. Indeed, if Alice uses a cipher that provides only semantic security (as in Definition 2.3), this may not provide Alice with any meaningful security guarantee, and may very well expose her to a realistic attack. For example, suppose Alice uses the stream cipher E discussed in Section 3.2. Here, Alice’s key is a seed s for a PRG G, and viewing a file m as a bit string, Alice encrypts m by computing the ciphertext c := m , where consists of the first m bits of the “key stream” G(s). But if Alice uses this same seed s to encrypt many files, an adversary can easily mount an attack. For example, if an adversary knows some of the bits of one file, he can directly compute the corresponding bits of the key stream, and hence obtain the corresponding bits of any file. How might an adversary know some bits of a given file? Well, certain files, like email messages, contain standard header information (see Example 2.6), and so if the adversary knows that a given ciphertext is an encryption of an email, he can get the bits of the key stream that correspond to the location of the bits in this 173
standard header. To mount an even more devastating attack, the adversary may try something even more devious: he could simply send Alice a large email, say one megabyte in length; assuming that Alice’s software automatically stores an encryption of this email on her server, when the adversary snoops her file server, he can recover a corresponding one megabyte chunk of the key stream, and now he decrypt any one megabyte file stored on Alice’s server! This email may even be caught in Alice’s spam filter, and never actually seen by Alice, although her encryption software may very well diligently encrypt this email along with everything else. This type of an attack is called a chosen plaintext attack, because the adversary forces Alice to give him the encryption of one or more plaintexts of his choice during his attack on the system. Clearly, the stream cipher above is inadequate for the job. In fact, the stream cipher, as well as any other deterministic cipher, should not be used to encrypt multiple files with the same key. Why? Any deterministic cipher that is used to encrypt several files with the same key will su↵er from an inherent weakness: an adversary will always be able to tell when two files are identical or not. Indeed, with a deterministic cipher, if the same key is used to encrypt the same message, the resulting ciphertext will always be the same (and conversely, for any cipher, if the same key is used to encrypt two di↵erent messages, the resulting ciphertexts must be di↵erent). While this type of attack is certainly not as dramatic as those discussed above, in which the adversary can read Alice’s files almost at will, it is still a serious vulnerability. For example, while the discussion in Section 4.1.4 about ECB mode was technically about encrypting a single message consisting of many data blocks, it applies equally well to the problem of encrypting many singleblock messages under the same key. In fact, it is possible for Alice to use a cipher to securely encrypt all of her files under a single, short key, but she will need to use a cipher that is better suited to this task. In particular, because of the above inherent weakness of any deterministic cipher, she will have to use a probabilistic cipher, that is, a cipher that uses a probabilistic encryption algorithm, so that di↵erent encryptions of the same plaintext under the same key will (generally) produce di↵erent encryptions. For her task, she will want a cipher that achieves a level of security stronger than semantic security. The appropriate notion of security is called semantic security against chosen plaintext attack. In Section 5.3 and the sections following, we formally define this concept, look at some constructions based on semantically secure ciphers, PRFs, and block ciphers, and look at a few case studies of “real world” systems. While the above discussion motivated the topics in this chapter using the example of the “file encryption” problem, one can also motivate these topics by considering the “secure network communication” problem. In this setting, one considers the situation where Alice and Bob share a secret key (or keys), and Alice wants to secretly transmit several of messages to Bob over an insecure network. Now, if Alice can conveniently concatenate all of her messages into one long message, then she can just use a stream cipher to encrypt the whole lot, and be done with it. However, for a variety of technical reasons, this may not be feasible: if she wants to be able to transmit the messages in an arbitrary order and at arbitrary times, then she is faced with a problem very similar to that of the “file encryption” problem. Again, if Alice and Bob want to use a single, short key, the right tool for the job is a cipher semantically secure against chosen plaintext attack. We stress again that just like in Chapter 2, the techniques covered in this chapter do not provide any data integrity, nor do they address the problem of how two parties come to share a secret key to begin with. These issues are dealt with in coming chapters.
174
5.2
Security against multikey attacks
Consider again the “file encryption” problem discussed in the introduction to this chapter. Suppose Alice chooses to encrypt each of her files under di↵erent, independently generated keys using a semantically secure cipher. Does semantic security imply a corresponding security property in this “multikey” setting? The answer to this question is “yes.” We begin by stating the natural security property corresponding to semantic security in the multikey setting. Attack Game 5.1 (multikey semantic security). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define Experiment b: • The adversary submits a sequence of queries to the challenger.
For i = 1, 2, . . . , the ith query is a pair of messages, mi0 , mi1 2 M, of the same length. The challenger computes ki
R
K, ci
R
E(ki , mib ), and sends ci to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to E as MSSadv[A, E] := Pr[W0 ] Pr[W1 ]. 2 We stress that in the above attack game, the adversary’s queries are adaptively chosen, in the sense that for each i = 1, 2, . . . , the message pair (mi0 , mi1 ) may be computed by the adversary in some way that depends somehow on the previous encryptions c1 , . . . , ci 1 output by the challenger. Definition 5.1 (Multikey semantic security). A cipher E is called multikey semantically secure if for all efficient adversaries A, the value MSSadv[A, E] is negligible. As discussed in Section 2.3.5, Attack Game 5.1 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A. In this game, we measure A’s bitguessing advantage MSSadv⇤ [A, E] as Pr[ˆb = b] 1/2, and as usual (by (2.13)), we have MSSadv[A, E] = 2 · MSSadv⇤ [A, E]. As the next theorem shows, semantic security implies multikey semantic security. Theorem 5.1. If a cipher E is semantically secure, it is also multikey semantically secure. In particular, for every MSS adversary A that attacks E as in Attack Game 5.1, and which makes at most Q queries to its challenger, there exists an SS adversary B that attacks E as in Attack Game 2.1, where B is an elementary wrapper around A, such that MSSadv[A, E] = Q · SSadv[B, E].
175
(5.1)
Proof idea. The proof is a straightforward hybrid argument, which is a proof technique we introduced in the proofs of Theorem 3.2 and 3.3 (the reader is advised to review those proofs, if necessary). In Experiment 0 of the MSS attack game, the challenger is encrypting m10 , m20 , . . . , mQ0 . Intuitively, since the key k1 is only used to encrypt the first message, and E is semantically secure, if we modify the challenger so that it encrypts m11 instead of m10 , the adversary should not behave significantly di↵erently. Similarly, we may modify the challenger so that it encrypts m21 instead of m20 , and the adversary should not notice the di↵erence. If we continue in this way, making a total of Q modifications to the challenger, we end up in Experiment 1 of the MSS game, and the adversary should not notice the di↵erence. 2 Proof. Suppose E = (E, D) is defined over (K, X , Y). Let A be an MSS adversary that plays Attack Game 5.1 with respect to E, and which makes at most Q queries to its challenger in that game. First, we introduce Q + 1 hybrid games, Hybrid 0, . . . , Hybrid Q, played between a challenger and A. For j = 0, 1, . . . , Q, when A makes its ith query (mi0 , mi1 ), the challenger in Hybrid j computes its response ci as follows: ki R K if i > j then
ci
R
E(ki , mi0 )
else
ci
R
E(ki , mi1 ).
Put another way, the challenger in Hybrid j encrypts m11 , . . . , mj1 ,
m(j+1)0 , . . . , mQ0 ,
generating di↵erent keys for each of these encryptions. For j = 0, 1, . . . , Q, let pj denote the probability that A outputs 1 in Hybrid j. Observe that p0 is equal to the probability that A outputs 1 in Experiment 0 of Attack Game 5.1 with respect to E, while pQ is equal to the probability that A outputs 1 in Experiment 1 of Attack Game 5.1 with respect to E. Therefore, we have MSSadv[A, E] = pQ
p0 .
(5.2)
We next devise an SS adversary B that plays Attack Game 2.1 with respect to E, as follows: First, B chooses ! 2 {1, . . . , Q} at random.
Then, B plays the role of challenger to A — when A makes its ith query (mi0 , mi1 ), B computes its response ci as follows: if i > ! then ki R K, ci R E(ki , mi0 ) else if i = ! then B submits (mi0 , mi1 ) to its own challenger ci is set to the challenger’s response else // i < ! ki R K, ci R E(ki , mi1 ). Finally, B outputs whatever A outputs. Put another way, adversary B encrypts m11 , . . . , m(! 176
1)1 ,
generating its own keys for this purpose, submits (m!0 , m!1 ) to its own encryption oracle, and encrypts m(!+1)0 , . . . , mQ0 , again, generating its own keys. We claim that MSSadv[A, E] = Q · SSadv[B, E].
(5.3)
To prove this claim, for b = 0, 1, let Wb be the event that B outputs 1 in Experiment b of its attack game. If ! denotes the random number chosen by B, then the key observation is that for j = 1, . . . , Q, we have: Pr[W0  ! = j] = pj
1
and
Pr[W1  ! = j] = pj .
Equation (5.3) now follows from this observation, together with (5.2), via the usual telescoping sum calculation: SSadv[B, E] = Pr[W1 ]
Pr[W0 ]
Q 1 X = · Pr[W1  ! = j] Q j=1
1 · pQ p0  Q 1 · MSSadv[A, E], = Q
Q X j=1
Pr[W0  ! = j]
=
and the claim, and hence the theorem, is proved. 2 Let us return now to the “file encryption” problem discussed in the introduction to this chapter. What this theorem says is that if Alice uses independent keys to encrypt each of her files with a semantically secure cipher, then an adversary who sees the ciphertexts stored on the file server will e↵ectively learn nothing about Alice’s files (except possibly some information about their lengths). Notice that this holds even if the adversary plays an active role in determining the contents of some of the files (e.g., by sending Alice an email, as discussed in the introduction).
5.3
Semantic security against chosen plaintext attack
Now we consider the problem that Alice faced in introduction of this chapter, where she wants to encrypt all of her files on her system using a single, and hopefully short, secret key. The right notion of security for this task is semantic security against chosen plaintext attack, or CPA security for short. Attack Game 5.2 (CPA security). For a given cipher E = (E, D), defined over (K, M, C), and for a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define Experiment b: • The challenger selects k
R
K. 177
• The adversary submits a sequence of queries to the challenger.
For i = 1, 2, . . . , the ith query is a pair of messages, mi0 , mi1 2 M, of the same length.
The challenger computes ci
R
E(k, mib ), and sends ci to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to E as CPAadv[A, E] := Pr[W0 ] Pr[W1 ]. 2 The only di↵erence between the CPA attack game and the MSS Attack Game 5.1 is that in the CPA game, the same key is used for all encryptions, whereas in the MSS attack game, a di↵erent key is chosen for each encryption. In particular, the adversary’s queries may adaptively chosen in the CPA game, just as in the MSS game. Definition 5.2 (CPA security). A cipher E is called semantically secure against chosen plaintext attack, or simply CPA secure, if for all efficient adversaries A, the value CPAadv[A, E] is negligible. As in Section 2.3.5, Attack Game 5.2 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A; we define A’s bitguessing advantage as CPAadv⇤ [A, E] := Pr[ˆb = b] 1/2, and as usual (by (2.13)), we have CPAadv[A, E] = 2 · CPAadv⇤ [A, E].
(5.4)
Again, we return to the “file encryption” problem discussed in the introduction to this chapter. What this definition says is that if Alice uses just a single key to encrypt each of her files with a CPA secure cipher, then an adversary who sees the ciphertexts stored on the file server will e↵ectively learn nothing about Alice’s files (except possibly some information about their lengths). Again, notice that this holds even if the adversary plays an active role in determining the contents of some of the files. Example 5.1. Just to exercise the definition a bit, let us show that no deterministic cipher can possibly satisfy the definition of CPA security. Suppose that E = (E, D) is a deterministic cipher. We construct a CPA adversary A as follows. Let m, m0 be any two, distinct messages in the message space of E. The adversary A makes two queries to its challenger: the first is (m, m0 ), and the second is (m, m). Suppose c1 is the challenger’s response to the first query and c2 is the challenger’s response to the second query. Adversary A outputs 1 if c1 = c2 , and 0 otherwise. Let us calculate CPAadv[A, E]. On then one hand, in Experiment 0 of Attack Game 5.2, the challenger encrypts m in responding to both queries, and so c1 = c2 ; hence, A outputs 1 with probability 1 in this experiment (this is precisely where we use the assumption that E is deterministic). On the other hand, in Experiment 1, the challenger encrypts m0 and m, and so c1 6= c2 ; hence, A outputs 1 with probability 0 in this experiment. It follows that CPAadv[A, E] = 1. The attack in this example can be generalized to show that not only must a CPAsecure cipher be probabilistic, but it must be very unlikely that two encryptions of the same message yield the same ciphertext — see Exercise 5.11. 2 Remark 5.1. Analogous to Theorem 5.1, it is straightforward to show that if a cipher is CPAsecure, it is also CPAsecure in the multikey setting. See Exercise 5.2. 2 178
5.4
Building CPA secure ciphers
In this section, we describe a number of ways of building ciphers that are semantically secure against chosen plaintext attack. As we have already discussed in Example 5.1, any such cipher must be probabilistic. We begin in Section 5.4.1 with a generic construction that combines any semantically secure cipher with a pseudorandom function (PRF). The PRF is used to generate “one time” keys. Next, in Section 5.4.2, we develop a probabilistic variant of the counter mode cipher discussed in Section 4.4.4. While this scheme can be based on any PRF, in practice, the PRF is usually instantiated with a block cipher. Finally, in Section 5.4.3, we present a cipher that is constructed from a block cipher using a method called cipher block chaining (CBC) mode. These last two constructions, counter mode and CBC mode, are called modes of operation of a block cipher. Another mode of operation we have already seen in Section 4.1.4 is electronic codebook (ECB) mode. However, because of the lack of security provided by this mode of operation, its is seldom used. There are other modes of operations that provide CPA security, which we develop in the exercises.
5.4.1
A generic hybrid construction
In this section, we show how to turn any semantically secure cipher E = (E, D) into a CPA secure cipher E 0 using an appropriate PRF F . The basic idea is this. A key for E 0 is a key k 0 for F . To encrypt a single message m, a random input x for F is chosen, and a key k for E is derived by computing k F (k 0 , x). Then m is R 0 encrypted using this key k: c E(k, m). The ciphertext is c := (x, c). Note that we need to include x as part of c0 so that we can decrypt: the decryption algorithm first derives the key k by D(k, c). computing k F (k 0 , x), and then recovers m by computing m For all of this to work, the output space of F must match the key space of E. Also, the input space of F must be superpoly, so that the chances of accidentally generating the same x value twice is negligible. Now the details. Let E = (E, D) be a cipher, defined over (K, M, C). Let F be a PRF defined over (K0 , X , K); that is, the output space of F should be equal to the key space of E. We define a new cipher E 0 = (E 0 , D0 ), defined over (K0 , M, X ⇥ C), as follows: • for k 0 2 K0 and m 2 M, we define E 0 (k 0 , m) :=
x R X, k F (k 0 , x), c output (x, c);
R
E(k, m)
• for k 0 2 K0 and c0 = (x, c) 2 X ⇥ C, we define D0 (k 0 , c0 ) :=
k F (k 0 , x), m output m.
D(k, c)
It is easy to verify that E 0 is indeed a cipher, and is our first example of a probabilistic cipher. Example 5.2. Before proving CPA security of E 0 let us first see the construction in action. Suppose E is the onetime pad, namely E(k, m) := k m where K = M = C = {0, 1}L . Applying the generic hybrid construction above to the onetime pad results in the following popular cipher E0 = (E0 , D0 ): • for k 0 2 K0 and m 2 M, define 179
E0 (k 0 , m) :=
x
R
X , output (x, F (k 0 , x)
m)
• for k 0 2 K0 and c0 = (x, c) 2 X ⇥ C, define D0 (k 0 , c0 ) := output F (k 0 , x)
c
CPA security of this cipher follows from the CPA security of the generic hybrid construction E 0 which is proved in Theorem 5.2 below. 2 Theorem 5.2. If F is a secure PRF, E is a semantically secure cipher, and N := X  is superpoly, then the cipher E 0 described above is a CPA secure cipher. In particular, for every CPA adversary A that attacks E 0 as in the bitguessing version of Attack Game 5.2, and which makes at most Q queries to its challenger, there exists a PRF adversary BF that attacks F as in Attack Game 4.2, and an SS adversary BE that attacks E as in the bitguessing version of Attack Game 2.1, where both BF and BE are elementary wrappers around A, such that Q2 + 2 · PRFadv[BF , F ] + Q · SSadv[BE , E]. (5.5) CPAadv[A, E 0 ] N
Proof idea. First, using the assumption that F is a PRF, we can e↵ectively replace F by a truly random function. Second, using the assumption that N is superpoly, we argue that except with negligible probability, no two xvalues are ever the same. But in this scenario, the challenger’s keys are now all independently generated, and so the challenger is really playing the same role as the challenger in the Attack Game 5.1. The result then follows from Theorem 5.1. 2 Proof. Let A be an efficient CPA adversary that attacks E 0 as in Attack Game 5.2. Assume that A makes at most Q queries to its challenger. Our goal is to show that CPAadv[A, E 0 ] is negligible, assuming that F is a secure PRF, that N is superpoly, and that E is semantically secure. It is convenient to use the bitguessing versions of the CPA and semantic security attack games. We prove: Q2 + PRFadv[BF , F ] + Q · SSadv⇤ [BE , E] (5.6) CPAadv⇤ [A, E 0 ] 2N for efficient adversaries BF and BE . Then (5.5) follows from (5.4) and Theorem 2.10. The basic strategy of the proof is as follows. First, we define Game 0 to be the game played between A and the challenger in the bitguessing version of Attack Game 5.2 with respect to E 0 . We then define several more games: Game 1, Game 2, and Game 3. Each of these games is played between A and a di↵erent challenger; moreover, as we shall see, Game 3 is equivalent to the bitguessing version of Attack Game 5.1 with respect to E. In each of these games, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, . . . , 3, we define Wj to be the event that ˆb = b in Game j. We will show that for j = 1, . . . , 3, the value Pr[Wj ] Pr[Wj 1 ] is negligible; moreover, from the assumption that E is semantically secure, and from Theorem 5.1, it will follow that Pr[W3 ] 1/2 is negligible; from this, it follows that CPAadv⇤ [A, E 0 ] := Pr[W0 ] 1/2 is negligible. Game 0. Let us begin by giving a detailed description of the challenger in Game 0 that is convenient for our purposes:
180
b R {0, 1} k 0 R K0 for i 1 to Q do xi R X F (k 0 , xi ) ki upon receiving the ith query (mi0 , mi1 ) 2 M2 : ci R E(ki , mib ) send (xi , ci ) to the adversary. By construction, we have CPAadv⇤ [A, E 0 ] = Pr[W0 ]
1/2 ,
(5.7)
Game 1. Next, we play our “PRF card,” replacing F (k 0 , ·) by a truly random function f 2 Funs[X , K]. The challenger in this game looks like this: b R {0, 1} f R Funs[X , K] for i 1 to Q do xi R X ki f (xi ) upon receiving the ith query (mi0 , mi1 ) 2 M2 : ci R E(ki , mib ) send (xi , ci ) to the adversary. We claim that Pr[W1 ]
Pr[W0 ] = PRFadv[BF , F ],
(5.8)
where BF is an efficient PRF adversary; moreover, since we are assuming that F is a secure PRF, it must be the case that PRFadv[BF , F ] is negligible. The design of BF is naturally suggested by the syntax of Games 0 and 1. If f 2 Funs[X , K] denotes the function chosen by its challenger in Attack Game 4.2 with respect to F , adversary BF runs as follows: First, BF makes the following computations: b R {0, 1} for i 1 to Q do xi R X ki R f (xi ). Here, BF obtains the value f (xi ) by querying its own challenger with xi .
Next, adversary BF plays the role of challenger to A; specifically, when A makes its ith query (mi0 , mi1 ), adversary BF computes ci
R
E(ki , mib )
and sends (xi , ci ) to A. 181
BF b PRF Challenger
xi ki
R
R
{0, 1}
A
mi0 , mi1
X ci
R
E(ki , mib ) xi , c i
ˆb
(ˆb, b)
Figure 5.1: Adversary BF in the proof of Theorem 5.2 Eventually, A halts and outputs a bit ˆb, at which time adversary BF halts and outputs 1 if ˆb = b, and outputs 0 otherwise. See Fig. 5.1 for a picture of adversary BF . As usual, (x, y) is defined to be 1 if x = y, and 0 otherwise. Game 2. Next, we use our “faithful gnome” idea (see Section 4.4.2) to implement the random function f . Our “gnome” has to keep track of the inputs to f , and detect if the same input is used twice. In the following logic, our gnome uses a truly random key as the “default” value for ki , but overrides this default value if necessary, as indicated in the line marked (⇤): b R {0, 1} for i 1 to Q do xi R X ki R K (⇤) if xi = xj for some j < i then ki
kj
upon receiving the ith query (mi0 , mi1 ) 2 M2 : ci R E(ki , mib ) send (xi , ci ) to the adversary. As this is a faithful implementation of the random function f , we have Pr[W2 ] = Pr[W1 ]. 182
(5.9)
Game 3. Next, we make our gnome “forgetful,” simply dropping the line marked (⇤) in the previous game: b R {0, 1} for i 1 to Q do xi R X ki R K upon receiving the ith query (mi0 , mi1 ) 2 M2 : ci R E(ki , mib ) send (xi , ci ) to the adversary. To analyze the quantity Pr[W3 ] Pr[W2 ], we use the Di↵erence Lemma (Theorem 4.7). To this end, we view Games 2 and 3 as operating on the same underlying probability space: the random choices made by the adversary and the challenger are identical in both games — all that di↵ers is the rule used by the challenger to compute its responses. In particular, the variables xi are identical in both games. Define Z to be the event that xi = xj for some i 6= j. Clearly, Games 2 and 3 proceed identically unless Z occurs; in particular, W2 ^ Z¯ occurs if and only if W3 ^ Z¯ occurs. Applying the Di↵erence Lemma, we therefore have Pr[W3 ]
Pr[W2 ] Pr[Z].
(5.10)
Q2 , 2N
(5.11)
Moreover, it is easy to see that Pr[Z]
since Z is the union of less that Q2 /2 events, each of which occurs with probability 1/N . Observe that in Game 3, independent encryption keys ki are used to encrypt each message. So next, we play our “semantic security card,” claiming that Pr[W3 ]
1/2 = MSSadv⇤ [B¯E , E],
(5.12)
where B¯E is an efficient adversary that plays the bitguessing version of Attack Game 5.1 with respect to E, making at most Q queries to its challenger in that game. The design of B¯E is naturally suggested by the syntactic form of Game 3. It works as follows: Playing the role of challenger to A, upon receiving the ith query (mi0 , mi1 ) from A, adversary B¯E submits (mi0 , mi1 ) to its own challenger, obtaining a ciphertext ci 2 C; then B¯E selects xi at random from X , and sends (xi , ci ) to A in response to the latter’s query. When A finally outputs a bit ˆb, B¯E outputs this same bit. See Fig. 5.2 for a picture of adversary B¯E . It is evident from the construction (and (2.13)) that (5.12) holds. Moreover, by Theorem 5.1 and (5.1), we have MSSadv⇤ [B¯E , E] = Q · SSadv⇤ [BE , E], (5.13) where BE is an efficient adversary playing the bitguessing version of Attack Game 2.1 with respect to E. 183
B¯E
MSS Challenger mi0 , mi1
A
mi0 , mi1
ci
xi
R
X xi , c i ˆb
Figure 5.2: Adversary B¯E in the proof of Theorem 5.2 Putting together (5.7) through (5.13), we obtain (5.6). Also, one can check that the running times of both BF and BE are roughly the same as that of A; indeed, they are elementary wrappers around A, and (5.5) holds regardless of whether A is efficient. 2 While the above proof was a bit long, we hope the reader agrees that it was in fact quite natural, and that all of the steps were fairly easy to follow. Also, this proof illustrates how one typically employs more than one security assumption in devising a security proof as a sequence of games. Remark 5.2. We briefly mention that the hybrid construction E 0 in Theorem 5.2 is CPA secure even if the PRF F used in the construction is only weakly secure (as in Definition 4.3). To prove Theorem 5.2 under this weaker assumption observe that in both Games 0 and 1 the challenger only evaluates the PRF at random points in X . Therefore, the adversary’s advantage in distinguishing Games 0 and 1 is negligible even if F is only weakly secure. 2
5.4.2
Randomized counter mode
We can build a CPA secure cipher directly out of a secure PRF, as follows. Suppose F is a PRF defined over (K, X , Y). We shall assume that X = {0, . . . , N 1}, and that Y = {0, 1}n . For any polybounded ` 1, we define a cipher E = (E, D), with key space K, message space ` Y , and ciphertext space X ⇥ Y ` , as follows: • for k 2 K and m 2 Y ` , with v := m, we define 184
E(k, m) := x R X compute c 2 Y v as follows: for j 0 to v 1 do c[j] F (k, x + j mod N ) output (x, c);
m[j]
• for k 2 K and c0 = (x, c) 2 X ⇥ Y ` , with v := c, we define D(k, c0 ) := compute m 2 Y v as follows: for j 0 to v 1 do m[j] F (k, x + j mod N ) output m.
c[j]
This cipher is much like the stream cipher one would get by building a PRG out of F using the construction in Section 4.4.4. The di↵erence is that instead of using a fixed sequence of inputs to F to derive a key stream, we use a random starting point, which we then increment to obtain successive inputs to F . The x component of the ciphertext is typically called an initial value, or IV for short. In practice, F is typically implemented using the encryption function of a block cipher, and X = Y = {0, 1}n , where we naturally view nbit strings as numbers in the range 0, . . . , 2n 1. As it happens, the decryption function of the block cipher is not needed at all in this construction. See Fig. 5.3 for an illustration of this mode. It is easy to verify that E is indeed a (probabilistic) cipher. Also, note that the message space of E is variable length, and that for the purposes of defining CPA security using Attack Game 5.2, the length of a message m 2 Y ` is its natural length m. Theorem 5.3. If F is a secure PRF and N is superpoly, then for any polybounded ` cipher E described above is a CPA secure cipher.
1, the
In particular, for every CPA adversary A that attacks E as in Attack Game 5.2, and which makes at most Q queries to its challenger, there exists a PRF adversary B that attacks F as in Attack Game 4.2, where B is an elementary wrapper around A, such that CPAadv[A, E]
4Q2 ` + 2 · PRFadv[B, F ]. N
(5.14)
Proof idea. Suppose we start with an adversary that plays the CPA attack game with respect to E. First, using the assumption that F is a PRF, we can e↵ectively replace F by a truly random function f . Second, using the assumption that N is superpoly, and the fact that each IV is chosen at random, we can argue that except with negligible probability, the challenger never evaluates f at the same point twice. But in this case, the challenger is e↵ectively encrypting each message using an independent onetime pad, and so we can conclude that the adversary’s advantage in the original CPA attack game is negligible. 2 Proof. Let A be an efficient adversary that plays Attack Game 5.2 with respect to E, and which makes at most Q queries to its challenger in that game. We want to show that CPAadv[A, E] is negligible, assuming that F is a secure PRF and that N is superpoly. 185
x
m[2]
m[1]
m[0] hx + 0in
hx + 1in
hx + 2in
E(k, ·)
E(k, ·)
E(k, ·)
c[0]
c[1]
c[2]
(a) encryption x
c[0]
c[1]
c[2]
hx + 0in
hx + 1in
hx + 2in
E(k, ·)
E(k, ·)
E(k, ·)
m[0]
m[1]
m[2]
(b) decryption
Figure 5.3: Randomizd counter mode (v = 3)
186
It is convenient to use the bitguessing version of the CPA attack game, We prove: 2Q2 ` + PRFadv[B, F ] (5.15) N for an efficient adversary B. Then (5.14) follows from (5.4). The basic strategy of the proof is as follows. First, we define Game 0 to be the game played between A and the challenger in the bitguessing version of Attack Game 5.2 with respect to E. We then define several more games: Game 1, Game 2, and Game 3. Each of these games is played between A and a di↵erent challenger. In each of these games, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, . . . , 3, we define Wj to be the event that ˆb = b in Game j. We will show that for j = 1, . . . , 3, the value Pr[Wj ] Pr[Wj 1 ] is negligible; moreover, it will be evident that Pr[W3 ] = 1/2, from which it will follow that CPAadv⇤ [A, E] := Pr[W0 ] 1/2 is negligible. CPAadv⇤ [A, E]
Game 0. We may describe the challenger in Game 0 as follows: b R {0, 1} k R K for i 1 to Q do xi R X for j 0 to ` 1 do xi + j mod N x0ij F (k, x0ij ) yij
upon receiving the ith query (mi0 , mi1 ), with vi := mi0  = mi1 : compute ci 2 Y vi as follows: yij mib [j] for j 0 to vi 1 do: ci [j] send (xi , ci ) to the adversary. By construction, we have we have CPAadv⇤ [A, E] = Pr[W0 ]
1/2 .
(5.16)
Game 1. Next, we play our “PRF card,” replacing F (k, ·) by a truly random function f 2 Funs[X , Y]. The challenger in this game looks like this: b R {0, 1} f R Funs[X , Y] for i 1 to Q do xi R X for j 0 to ` 1 do x0ij xi + j mod N f (x0ij ) yij ···
We have left out part of the code for the challenger, as it will not change in any of our games. We claim that (5.17) Pr[W1 ] Pr[W0 ] = PRFadv[B, F ], where B is an efficient adversary; moreover, since we are assuming that F is a secure PRF, it must be the case that PRFadv[B, F ] is negligible. This is hopefully (by now) a routine argument, and we leave the details of this to the reader. 187
Game 2. Next, we use our “faithful gnome” idea to implement the random function f . In describing the logic of our challenger in this game, we use the standard lexicographic ordering on pairs of indices (i, j); that is, (i0 , j 0 ) < (i, j) if and only if i0 < i
or
i0 = i and j 0 < j.
In the following logic, our “gnome” uses a truly random value as the “default” value for each yij , but overrides this default value if necessary, as indicated in the line marked (⇤): b R {0, 1} for i 1 to Q do xi R X for j 0 to ` 1 do 0 xi + j mod N xij yij R Y (⇤) if x0ij = x0i0 j 0 for some (i0 , j 0 ) < (i, j) then yij
yi0 j 0
··· As this is a faithful implementation of the random function f , we have Pr[W2 ] = Pr[W1 ].
(5.18)
Game 3. Now we make our gnome “forgetful,” dropping the line marked (⇤) in the previous game: b R {0, 1} for i 1 to Q do xi R X for j 0 to ` 1 do x0ij xi + j mod N R yij Y ···
To analyze the quantity Pr[W3 ] Pr[W2 ], we use the Di↵erence Lemma (Theorem 4.7). To this end, we view Games 2 and 3 as operating on the same underlying probability space: the random choices made by the adversary and the challenger are identical in both games — all that di↵ers is the rule used by the challenger to compute its responses. In particular, the variables x0ij are identical in both games. Define Z to be the event that x0ij = x0i0 j 0 for some (i, j) 6= (i0 , j 0 ). Clearly, Games 2 and 3 proceed identically unless Z occurs; in particular, W2 ^ Z¯ occurs if and only if W3 ^ Z¯ occurs. Applying the Di↵erence Lemma, we therefore have Pr[W3 ]
Pr[W2 ] Pr[Z].
(5.19)
We claim that
2Q2 ` . (5.20) N To prove this claim, we may assume that N 2` (this should anyway generally hold, since we are assuming that ` is polybounded and N is superpoly). Observe that Z occurs if and only if Pr[Z]
{xi , . . . , xi + `
1} \ {xi0 , . . . , xi0 + ` 188
1} 6= ;
for some pair of indices i and i0 with i 6= i0 (and arithmetic is done mod N ). Consider any fixed such pair of indices. Conditioned on any fixed value of xi , the value xi0 is uniformly distributed over {0, . . . , N 1}, and the intervals overlap if and only if xi0 2 {xi + j : which happens with probability (2`
`+1j `
1},
1)/N . The inequality (5.20) now follows.
Finally, observe that in Game 3 the yij values are uniformly and independently distributed over Y, and thus the challenger is essentially using independent onetime pads to encrypt. In particular, it is easy to see that the adversary’s output in this game is independent of b. Therefore, Pr[W3 ] = 1/2.
(5.21)
Putting together (5.16) through (5.21), we obtain (5.15), and the theorem follows. 2 Remark 5.3. One can also view randomized counter mode as a special case of the generic hybrid construction in Section 5.4.1. See Exercise 5.5. 2 Case study: AES counter mode The IPsec protocol uses a particular variants of AES counter mode, as specified in RFC 3686. Recall that AES uses a 128 bit block. Rather than picking a random 128bit IV for every message, RFC 3686 picks the IV as follows: • The most significant 32 bits are chosen at random at the time that the secret key is generated and are fixed for the life of the key. The same 32 bit value is used for all messages encrypted using this key. • The next 64 bits are chosen at random in {0, 1}64 . • The least significant 32 bits are set to the number 1. This resulting 128bit IV is used as the initial value of the counter. When encryption a message the least significant 32 bits are incremented by one for every block of the message. Consequently, the maximum message length that can be encrypted is 232 AES blocks or 236 bytes. With this choice of IV the decryptor knows the 32 most significant bits of the IV as well as the 32 least significant bits. Hence, only 64 bits of the IV need to be sent with the ciphertext. The proof of Theorem 5.3 can be adapted to show that this method of choosing IVs is secure. The slight advantage of this method over picking a random 128bit IV is that the resulting ciphertext is a little shorter. A random IV forces the encryptor to include all 128 bits in the ciphertext. With the method of RFC 3686 only 64 bits are needed, thus shrinking the ciphertext by 8 bytes.
5.4.3
CBC mode
An historically important encryption method is to use a block cipher in cipher block chaining (CBC) mode. This method is used in older versions of the TLS protocol (e.g., TLS 1.0). It is inferior to counter mode encryption as discussed in the next section. Suppose E = (E, D) is a block cipher defined over (K, X ), where X = {0, 1}n . Let N := X  = n 1, we define a cipher E 0 = (E 0 , D0 ), with key space K, message 2 . For any polybounded ` ` space X , and ciphertext space X `+1 \ X 0 ; that is, the ciphertext space consists of all nonempty sequences of at most ` + 1 data blocks. Encryption and decryption are defined as follows: 189
• for k 2 K and m 2 X ` , with v := m, we define E 0 (k, m) := compute c 2 X v+1 as follows: c[0] R X for j 0 to v 1 do c[j + 1] E(k, c[j] output c;
m[j])
• for k 2 K and c 2 X `+1 \ X 0 , with v := c D0 (k, c) := compute m 2 X v as follows: for j 0 to v 1 do m[j] D(k, c[j + 1]) output m.
1, we define
c[j]
See Fig. 5.4 for an illustration of the encryption and decryption algorithm in the case m = 3. Here, the first component c[0] of the ciphertext is also called an initial value, or IV. Note that unlike the counter mode construction in Section 5.4.2, in CBC mode, we must use a block cipher, as we actually need to use the decryption algorithm of the block cipher. It is easy to verify that E 0 is indeed a (probabilistic) cipher. Also, note that the message space of E is variable length, and that for the purposes of defining CPA security using Attack Game 5.2, the length of a message m 2 X ` is its natural length m. Theorem 5.4. If E = (E, D) is a secure block cipher defined over (K, X ), and N := X  is superpoly, then for any polybounded ` 1, the cipher E 0 described above is a CPA secure cipher. In particular, for every CPA adversary A that attacks E 0 as in the bitguessing version of Attack Game 5.2, and which makes at most Q queries to its challenger, there exists BC adversary B that attacks E as in Attack Game 4.1, where B is an elementary wrapper around A, such that CPAadv[A, E 0 ]
2Q2 `2 + 2 · BCadv[B, E]. N
(5.22)
Proof idea. The basic idea of the proof is very similar to that of Theorem 5.3. We start with an adversary that plays the CPA attack game with respect to E 0 . We then replace E by a truly random function f . Then we argue that except with negligible probability, the challenger never evaluates f at the same point twice. But then what the adversary sees is nothing but a bunch of random bits, and so learns nothing at all about the message being encrypted. 2 Proof. Let A be an efficient CPA adversary that attacks E 0 as in Attack Game 5.2. Assume that A makes at most Q queries to its challenger in that game. We want to show that CPAadv⇤ [A, E 0 ] is negligible, assuming that E is a secure block cipher and that N is superpoly. Under these assumptions, by Corollary 4.5, the encryption function E is a secure PRF, defined over (K, X , X ). It is convenient to use the bitguessing version of the CPA attack game, We prove: CPAadv⇤ [A, E 0 ]
Q2 ` 2 + BCadv[B, E] N
for an efficient adversary B. Then (5.22) follows from (5.4). 190
(5.23)
m[0]
c[0]
m[1]
m[2]
E(k, ·)
E(k, ·)
E(k, ·)
c[1]
c[2]
c[3]
(a) encryption c[0]
c[1]
c[2]
c[3]
D(k, ·)
D(k, ·)
D(k, ·)
m[0]
m[1]
m[2]
(b) decryption Figure 5.4: Encryption and decryption for CBC mode with ` = 3
191
As usual, we define a sequence of games: Game 0, Game 1, Game 2, Game 3. Each of these games is played between A and a challenger. The challenger in Game 0 is the one from the bitguessing version of Attack Game 5.2 with respect to E 0 . In each of these games, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, . . . , 3, we define Wj to be the event that ˆb = b in Game j. We will show that for j = 1, . . . , 3, the value Pr[Wj ] Pr[Wj 1 ] is negligible; moreover, it will be evident that Pr[W3 ] = 1/2, from which it will follow that Pr[W0 ] 1/2 is negligible. Here we go! Game 0. We may describe the challenger in Game 0 as follows: b
R
{0, 1}, k
R
K
upon receiving the ith query (mi0 , mi1 ), with vi := mi0  = mi1 : compute ci 2 X vi +1 as follows: ci [0] R X for j 0 to vi 1 do ci [j] mib [j] xij ci [j + 1] E(k, xij ) send ci to the adversary. By construction, we have CPAadv⇤ [A, E 0 ] = Pr[W0 ]
1/2 .
(5.24)
Game 1. We now play the “PRF card,” replacing E(k, ·) by a truly random function f 2 Funs[X , X ]. Our challenger in this game looks like this: b
R
{0, 1}, f
R
Funs[X , X ]
upon receiving the ith query (mi0 , mi1 ), with vi := mi0  = mi1 : compute ci 2 X vi +1 as follows: ci [0] R X for j 0 to vi 1 do ci [j] mib [j] xij f (xij ) ci [j + 1] send ci to the adversary. We claim that Pr[W1 ]
Pr[W0 ] = PRFadv[B, E],
(5.25)
where B is an efficient adversary; moreover, since we are assuming that E is a secure block cipher, and that N is superpoly, it must be the case that PRFadv[B, E] is negligible. This is hopefully (by now) a routine argument, and we leave the details of this to the reader. Game 2. The next step in this dance should by now be familiar: we implement f using a faithful gnome. We do so by introducing random variables yij which represent the “default” values for ci [j], which get overridden if necessary in the line marked (⇤) below:
192
b R {0, 1} set yij R X for i = 1, . . . , Q and j = 0, . . . , `
upon receiving the ith query (mi0 , mi1 ), with vi := mi0  = mi1 : compute ci 2 X vi +1 as follows: yi0 ci [0] for j 0 to vi 1 do ci [j] mib [j] xij yi(j+1) ci [j + 1] (⇤) if xij = xi0 j 0 for some (i0 , j 0 ) < (i, j) then ci [j + 1] send ci to the adversary.
ci0 [j 0 + 1]
We clearly have Pr[W2 ] = Pr[W1 ].
(5.26)
Game 3. Now we make gnome forgetful, removing the check in the line marked (⇤): b R {0, 1} set yij R X for i = 1, . . . , Q and j = 0, . . . , `
upon receiving the ith query (mi0 , mi1 ), with vi := mi0  = mi1 : compute ci 2 X vi +1 as follows: yi0 ci [0] for j 0 to vi 1 do xij ci [j] mib [j] ci [j + 1] R yi(j+1) send ci to the adversary.
To analyze the quantity Pr[W3 ] Pr[W2 ], we use the Di↵erence Lemma (Theorem 4.7). To this end, we view Games 2 and 3 as operating on the same underlying probability space: the random choices made by the adversary and the challenger are identical in both games — all that di↵ers is the rule used by the challenger to compute its responses. We define Z to be the event that xij = xi0 j 0 in Game 3. Note that the event Z is defined in terms of the xij values in Game 3. Indeed, the xij values may not be computed in the same way in Games 2 and 3, and so we have explicitly defined the event Z in terms of their values in Game 3. Nevertheless, it is clear that Games 2 and 3 proceed identically unless Z occurs; in particular, W2 ^ Z¯ occurs if and only if W3 ^ Z¯ occurs. Applying the Di↵erence Lemma, we therefore have Pr[W3 ]
Pr[W2 ] Pr[Z].
(5.27)
We claim that
Q 2 `2 . (5.28) 2N To prove this, let Coins denote the random choices made by A. Observe that in Game 3, the values Pr[Z]
Coins, b, yij (i = 1, . . . Q, j = 0, . . . , `) are independently distributed. Consider any fixed index i = 1, . . . , Q. Let us condition on any fixed values of Coins, b, and yi0 j for i0 = 1, . . . , i 1 and j = 0, . . . , `. In this conditional probability space, the values of 193
mi0 , mi1 , and vi are completely determined, as are the values vi0 and xi0 j for i0 = 1, . . . , i 1 and j = 0, . . . , vi0 1; however, the values of yi0 , . . . , yi` are still uniformly and independently distributed over X . Moreover, as xij = yij mib [j] for j = 0, . . . , vi 1, it follows that these xij values are also uniformly and independently distributed over X . Thus, for any fixed index j = 0, . . . , vi 1, and any fixed indices i0 and j 0 , with (i0 , j 0 ) < (i, j), the probability that xij = xi0 j 0 in this conditional probability space is 1/N . The bound (5.28) now follows from an easy calculation. Finally, we claim that Pr[W3 ] = 1/2.
(5.29)
This follows from the fact that Coins, b, yij (i = 1, . . . Q, j = 0, . . . , `) are independently distributed, and the fact that the adversary’s output ˆb is a function of Coins, yij (i = 1, . . . Q, j = 0, . . . , `). From this, we see that ˆb and b are independent, and so (5.29) follows immediately. Putting together (5.24) through (5.29), we have CPAadv⇤ [A, E 0 ]
Q 2 `2 + PRFadv[B, E]. 2N
By Theorem 4.4, we have BCadv[B, E]
PRFadv[B, E]
Q 2 `2 , 2N
and (5.23) follows, which proves the theorem. 2
5.4.4
Case study: CBC padding in TLS 1.0
Let E = (E, D) be a block cipher with domain X . Our description of CBC mode encryption using E assumes that messages to be encrypted are elements of X ` . When the domain is X = {0, 1}128 , as in the case of AES, this implies that the length of messages to be encrypted must be a multiple of 16 bytes. Since the length of messages in practice need not be a multiple of 16 we need a way augment CBC to handle messages whose length is not necessarily a multiple of the block size. Suppose we wish to encrypt a vbyte message m using AES in CBC mode when v is not necessarily a multiple of 16. The first thing that comes to mind is to somehow pad the message m so that its length in bytes is a multiple of 16. Clearly the padding function needs to be invertible so that during decryption the padding can be removed. The TLS 1.0 protocol defines the following padding function for encrypting a vbyte message with AES in CBC mode: let p := 16 (v mod 16), then append p bytes to the message m where the content of each byte is value p 1. For example, consider the following two cases: • if m is 29 bytes long then p = 3 and the pad consists of the three bytes “222” so that the padded message is 32 bytes long which is exactly two AES blocks. • if the length of m is a multiple of the block size, say 32 bytes, then p = 16 and the pad consists of 16 bytes. The padded message is then 48 bytes long which is three AES blocks. 194
It may seem odd that when the message is a multiple of the block size we add a full dummy block at the end. This is necessary so that the decryption procedure can properly remove the pad. Indeed, it should be clear that this padding method is invertible for all input message lengths. It is an easy fact to prove that every invertible padding scheme for CBC mode encryption built from a secure block cipher gives a CPA secure cipher for messages of arbitrary length. Padding in CBC mode can be avoided using a method called ciphertext stealing as long as the plaintext is longer than a single block. The ciphertext stealing variant of CBC is the topic of Exercise 5.16. When encrypting messages whose length is less than a block, say single byte messages, there is still a need to pad.
5.4.5
Concrete parameters and a comparison of counter and CBC modes
We conclude this section with a comparison of the counter and CBC mode constructions. We assume that counter mode is implemented with a PRF F that maps nbit blocks to nbit blocks, and that CBC is implemented with an nbit block cipher. In each case, the message space consists of sequences of at most ` nbit data blocks. With the security theorems proved in this section, we have the following bounds: 4Q2 ` + 2 · PRFadv[BF , F ], 2n 2Q2 `2 + 2 · BCadv[BE , E]. CPAadv[A, Ecbc ] 2n CPAadv[A, Ectr ]
Here, A is any CPA adversary making at most Q queries to its challenger, ` is the maximum length (in data blocks) of any one message. For the purposes of this discussion, let us simply ignore the terms PRFadv[BF , F ] and BCadv[BE , E]. One can immediately see that counter mode has a quantitative security advantage. To make things more concrete, suppose the block size is n = 128, and that each message is 1MB (223 bits) so that ` = 216 blocks. If we want to keep the adversary’s advantage below 2 32 , then for counter mode, we can encrypt up to Q = 239.5 messages, while for CBC we can encrypt only up to 232 messages. Once Q message are encrypted with a given key, a fresh key must be generated and used for subsequent messages. Therefore, with counter mode a single key can be used to securely encrypt many more messages as compared with CBC. Counter mode has several other advantages over CBC: • Parallelism and pipelining. Encryption and decryption for counter mode is trivial to parallelize, whereas encryption in CBC mode is inherently sequential (decryption in CBC mode is parallelizable). Modes that support parallelism greatly improve performance when the underlying hardware can execute many instructions in parallel as is often the case in modern processors. More importantly, consider a hardware implementation of a single block cipher round that supports pipelining, as in Intel’s implementation of AES128 (page 118). Pipelining enables multiple encryption instructions to execute at the same time. A parallel mode such as counter mode keeps the pipeline busy, where as in CBC encryption the pipeline is mostly unused due to the sequential nature of this mode. As a result, counter mode encryption on Intel’s Haswell processors is about seven times faster than CBC mode encryption, assuming the plaintext data is already loaded into L1 cache. 195
• Shorter ciphertext length. For very short messages, counter mode ciphertexts are significantly shorter than CBC mode ciphertexts. Consider, for example, a onebyte plaintext (which arises naturally when encrypting individual key strokes as in SSH). A counter mode ciphertext need only be one block plus one byte: one block for the random IV plus one byte for the encrypted plaintext. In contrast, a CBC ciphertext is two full blocks. This results in 15 redundant bytes per CBC ciphertext assuming 128bit blocks. • Encryption only. CBC mode uses both algorithms E and D of the block cipher where as counter mode uses only algorithm E. This can reduce an implementation code size. Remark 5.4. Both randomized counter mode and CBC require a random IV. Some crypto libraries actually leave it to the higherlevel application to supply the IV. This can lead to problems if the higherlevel applications do not take pains to ensure the IVs are sufficiently random. For example, for counter mode, it is necessary that the IVs are sufficiently spread out, so that the corresponding intervals do not overlap. In fact, this property is sufficient as well. In contrast, for CBC mode, more is required: it is essential that IVs be unpredictable — see Exercise 5.12. Leaving it to the higherlevel application to supply the IV is actually an example of noncebased encryption, which we will explore in detail next, in Section 5.5. 2
5.5
Noncebased encryption
All of the CPAsecure encryption schemes we have seen so far su↵er from ciphertext expansion: ciphertexts are longer than plaintexts. For example, the generic hybrid construction in Section 5.4.1 generates ciphertexts (x, c), where x belongs to the input space of some PRF and c encrypts the actual message; the counter mode construction in Section 5.4.2 generates ciphertexts of the essentially same form (x, c); similarly, the CBC mode construction in Section 5.4.3 includes the IV as a part of the ciphertext. For very long messages, the expansion is not too bad. For example, with AES and counter mode or CBC mode, a 1MB message results is a ciphertext that is just 16 bytes longer, which may be a perfectly acceptable expansion rate. However, for messages of 16 bytes or less, ciphertexts are at least twice as long as plaintexts. The bad news is, some amount of ciphertext expansion is inevitable for any CPAsecure encryption scheme (see Exercise 5.10). The good news is, in certain settings, one can get by without any ciphertext expansion. For example, suppose Alice and Bob are fully synchronized, so that Alice first sends an encryption m1 , then an encryption m2 , and so on, while Bob first decrypts the encryption of m1 , and then decrypts the encryption of m2 , and so on. For concreteness, assume Alice and Bob are using the generic hybrid construction of Section 5.4.1. Recall that the encryption of message mi is (xi , ci ), where ci := E(ki , mi ) and ki := F (xi ). The essential property of the xi ’s needed to ensure security was simply that they are distinct. When Alice and Bob are fully synchronized (i.e., ciphertexts sent by Alice reach Bob inorder), they simply have to agree on a fixed sequence x1 , x2 , . . . , of distinct elements in the input space of the PRF F . For example, xi might simply be the binary encoding of i. This mode of operation of an encryption scheme does not really fit into our definitional framework. Historically, there are two ways to modify the framework to allow for this type of operation. One approach is to allow for stateful encryption schemes, where both the encryption and decryption algorithms maintain some internal state that evolves with each application of the algorithm. In the 196
example of the previous paragraph, the state would just consist of a counter that is incremented with each application of the algorithm. This approach requires encryptor and decryptor to be fully synchronized, which limits its applicability, and we shall not discuss it further. The second, and more popular, approach is called noncebased encryption. Instead of maintaining internal states, both the encryption and decryption algorithms take an additional input N , called a nonce. The syntax for noncebased encryption becomes c = E(k, m, N ), where c 2 C is the ciphertext, k 2 K is the key, m 2 M is the message, and N 2 N is the nonce. Moreover, the encryption algorithm E is required to be deterministic. Likewise, the decryption syntax becomes m = D(k, c, N ). The intention is that a message encrypted with a particular nonce should be decrypted with the same nonce — it is up to the application using the encryption scheme to enforce this. More formally, the correctness requirement is that D(k, E(k, m, N ),
N)
=m
for all k 2 K, m 2 M, and N 2 N . We say that such a noncebased cipher E = (E, D) is defined over (K, M, C, N ). Intuitively, a noncebased encryption scheme is CPA secure if it does not leak any useful information to an eavesdropper, assuming that no nonce is used more than once in the encryption process — again, it is up to the application using the scheme to enforce this. Note that this requirement on how nonces are used is very weak, much weaker than requiring that they are unpredictable, let alone randomly chosen. We can readily formalize this notion of security by slightly tweaking our original definition of CPA security. Attack Game 5.3 (noncebased CPA security). For a given cipher E = (E, D), defined over (K, M, C, N ), and for a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define Experiment b: • The challenger selects k
R
K.
• The adversary submits a sequence of queries to the challenger.
For i = 1, 2, . . . , the ith query is a pair of messages, mi0 , mi1 2 M, of the same length, and a nonce N i 2 N \ {N 1 , . . . , N i 1 }. The challenger computes ci
E(k, mib , N i ), and sends ci to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to E as nCPAadv[A, E] := Pr[W0 ] Pr[W1 ]. 2 197
Note that in the above game, the nonces are completely under the adversary’s control, subject only to the constraint that they are unique. Definition 5.3 (noncebased CPA security). A noncebased cipher E is called semantically secure against chosen plaintext attack, or simply CPA secure, if for all efficient adversaries A, the value nCPAadv[A, E] is negligible. As usual, as in Section 2.3.5, Attack Game 5.3 can be recast as a “bit guessing” game, and we have nCPAadv[A, E] = 2 · nCPAadv⇤ [A, E], (5.30)
where nCPAadv⇤ [A, E] := Pr[ˆb = b] just chooses b at random.
5.5.1
1/2 in a version of Attack Game 5.3 where the challenger
Noncebased generic hybrid encryption
Let us recast the generic hybrid construction in Section 5.4.1 as a noncebased encryption scheme. As in that section, E is a cipher, which we shall now insist is deterministic, defined over (K, M, C), and F is a PRF defined over (K0 , X , K). We define the noncebased cipher E 0 , which is defined over (K0 , M, C, X ), as follows: • for k 0 2 K0 , m 2 M, and x 2 X , we define E 0 (k 0 , m, x) := E(k, m), where k := F (k0 , x); • for k 0 2 K0 , c 2 C, x 2 X , we define D0 (k 0 , c) := D(k, c), where k := F (k0 , x). All we have done is to treat the value x 2 X as a nonce; otherwise, the scheme is exactly the same as that defined in Section 5.4.1. One can easily verify the correctness requirement for E 0 . Moreover, one can easily adapt the proof of Theorem 5.2 to prove that the following: Theorem 5.5. If F is a secure PRF and E is a semantically secure cipher, then the cipher E 0 described above is a CPA secure cipher. In particular, for every nCPA adversary A that attacks E 0 as in the bitguessing version of Attack Game 5.3, and which makes at most Q queries to its challenger, there exists a PRF adversary BF that attacks F as in Attack Game 4.2, and an SS adversary BE that attacks E as in the bitguessing version of Attack Game 2.1, where both BF and BE are elementary wrappers around A, such that nCPAadv[A, E 0 ] 2 · PRFadv[BF , F ] + Q · SSadv[BE , E].
(5.31)
2
We leave the proof as an exercise for the reader. Note that the term QN in (5.5), which represent the probability of a collision on the input to F , is missing from (5.31), simply because by definition, no collisions can occur.
5.5.2
Noncebased Counter mode
Next, we recast the countermode cipher from Section 5.4.2 to the noncebased encryption setting. Let us make a first attempt, by simply treating the value x 2 X in that construction as a nonce. Unfortunately, this scheme cannot satisfy the definition of noncebased CPA security. The problem is, an attacker could choose two distinct nonces x1 , x2 2 X , such that the intervals 198
{x1 , . . . , x1 + ` 1} and {x2 , . . . , x2 + ` 1} overlap (again, arithmetic is done mod N ). In this case, the security proof will break down; indeed, it is easy to mount a quite devastating attack, as discussed in Section 5.1, since that attacker can essentially force the encryptor to reuse some of the same bits of the “key stream”. Fortunately, the fix is easy. Let us assume that ` divides N (in practice, both ` and N will be powers of 2, so this is not an issue). Then we use as the nonce space {0, . . . , N/` 1}, and translate the nonce N to the PRF input x := N `. It is easy to see that for any two distinct nonces N 1 and N 2 , for x1 := N 1 ` and x2 := N 2 `, the intervals {x1 , . . . , x1 + ` 1} and {x2 , . . . , x2 + ` 1} do not overlap. With E modified in this way, we can easily adapt the proof of Theorem 5.3 to prove the following: Theorem 5.6. If F is a secure PRF, then the noncebased cipher E described above is CPA secure. In particular, for every nCPA adversary A that attacks E as in Attack Game 5.3, there exists a PRF adversary B that attacks F as in Attack Game 4.2, where B is an elementary wrapper around A, such that nCPAadv[A, E] 2 · PRFadv[B, F ]. (5.32)
We again leave the proof as an exercise for the reader.
5.5.3
Noncebased CBC mode
Finally, we consider how to recast the CBCmode encryption scheme in Section 5.4.3 as a noncebased encryption scheme. As a first attempt, one might simply try to view the IV c[0] as a nonce. Unfortunately, this does not yield a CPA secure noncebased encryption scheme. In the nCPA attack game, the adversary could make two queries: (m10 , m11 , N 1 ), (m20 , m21 , N 2 ), where m10 = N 1 6= N 2 = m20 , m11 = m21 . Here, all messages are oneblock messages. In Experiment 0 of the attack game, the resulting ciphertexts will be the same, whereas in Experiment 1, they will be di↵erent. Thus, we can perfectly distinguish between the two experiments. Again, the fix is fairly straightforward. The idea is to map nonces to pseudorandom IV’s by passing them through a PRF. So let us assume that we have a PRF F defined over (K0 , N , X ). Here, the key space K0 and input space N of F may be arbitrary sets, but the output space X of F must match the block space of the underlying block cipher E = (E, D), which is defined over (K, X ). In the noncebased CBC scheme E 0 , the key space is K ⇥ K0 , and in the encryption and decryption algorithms, the IV is computed from the nonce N and key k 0 as c[0] := F (k 0 , N ). With these modifications, we can now prove the following variant of Theorem 5.4: Theorem 5.7. If E = (E, D) is a secure block cipher defined over (K, X ), and N := X  is superpoly, and F is a secure PRF defined over (K0 , N , X ), then for any polybounded ` 1, the noncebased cipher E 0 described above is CPA secure.
199
In particular, for every nCPA adversary A that attacks E 0 as in the bitguessing version of Attack Game 5.3, and which makes at most Q queries to its challenger, there exists BC adversary B that attacks E as in Attack Game 4.1, and a PRF adversary BF that attacks F as in Attack Game 4.2, where B and BF are elementary wrappers around A, such that nCPAadv[A, E 0 ]
2Q2 `2 + 2 · PRFadv[BF , F ] + 2 · BCadv[B, E]. N
(5.33)
Again, we leave the proof as an exercise for the reader. Note that in the above construction, we may use the underlying block cipher E for the PRF F ; however, it is essential that independent keys k and k 0 are used (see Exercise 5.14).
5.6
A fun application: revocable broadcast encryption
Movie studios spend a lot of e↵ort making blockbuster movies, and then sell the movies (on DVDs) to millions of customers who purchase them to watch at home. A customer should be able to watch movies on a stateless standalone movie player, that has no network connection. The studios are worried about piracy, and do not want to send copyrighted digital content in the clear to millions of users. A simple solution could work as follows. Every authorized manufacturer is given a device key kd 2 K, and it embeds this key in every device that it sells. If there are a (1) (100) hundred authorized device manufacturers, then there are a hundred device keys kd , . . . , kd . A movie m is encrypted as: 9 8 k R K > > > > = < (i) R E(k , k) for i = 1, . . . , 100 : c i d cm := > > c R E 0 (k, m) > > ; : output (c1 , . . . , c100 , c)
where (E, D) is a CPA secure cipher, and (E 0 , D0 ) is semantically secure with key space K. We analyze this construction in Exercise 5.4, where we show that it is CPA secure. We refer to (c1 , . . . , c100 ) as the ciphertext header, and refer to c is the body. Now, every authorized device can decrypt the movie using its embedded device key. First, decrypt the appropriate ciphertext in the header, and then use the obtained key k to decrypt the body. This mechanism forms the basis of the content scrambling system (CSS) used to encrypted DVDs. We previously encountered CSS in Section 3.8. The trouble with this scheme is that once a single device is comprised, and its device key kd is extracted and published, then anyone can use this kd to decrypt every movie ever published. There is no way to revoke kd without breaking many consumer devices in the field. In fact, this is exactly how CSS was broken: the device key was extracted from an authorized player, and then used in a system called DeCSS to decrypt encrypted DVDs. The lesson from CSS is that global unrevocable device keys are a bad idea. Once a single key is leaked, all security is lost. When the DVD format was updated to a new format called Bluray, the industry got a second chance to design the encryption scheme. In the new scheme, called the Advanced Access Content System (AACS), every device gets a random device key unique to that device. The system is designed to support billions of devices, each with its own key. The goals of the system are twofold. First, every authorized device should be able to decrypt every Bluray disk. Second, whenever a device key is extracted and published, it should be possible 200
k15 k13
k14
k9 k1
k10 k2
k3
k11 k4
k5
k12 k6
k7
k8
Figure 5.5: The tree of keys for n = 8 devices; shaded nodes are the keys embedded in device 3. to revoke that key, so that this device key cannot be used to decrypt future Bluray disks, but without impacting any other devices in the field. A revocable broadcast system. Suppose there are n devices in the system, where for simplicity, let us assume n is a power of two. We treat these n devices as the leaves of a complete binary tree, as shown in Fig. 5.5. Every internal node in the tree is assigned a random key in the key space K. The keys embedded in device number i 2 {1, . . . , n} is the set of keys on the path from leaf number i to the root. This way, every device is given exactly log2 n keys in K. When the system is first launched, and no device keys are yet revoked, all content is encrypted using the key at the root (key number 15 in Fig. 5.5). More precisely, we encrypt a movie m as: cm :=
k
R
K, c1
R
E(kroot , k), c
R
E 0 (k, m), output (c1 , c)
Because all devices have the root key kroot , all devices can decrypt. Revoking devices. Now, suppose device number i is attacked, and all the keys stored on it are published. Then all future content will be encrypted using the keys associated with the siblings of the log2 n nodes on the path from leaf i to the root. For example, when device number 3 in Fig. 5.5 is revoked, all future content is encrypted using the three keys k4 , k9 , k14 as 8 9 k R K > > > > < = c1 R E(k4 , k), c2 R E(k9 , k), c3 R E(k14 , k) cm := (5.34) c R E 0 (k, m) > > > > : ; output (c1 , c2 , c3 , c)
Again, (c1 , c2 , c3 ) is the ciphertext header, and c is the ciphertext body. Observe that device number 3 cannot decrypt cm , because it cannot decrypt any of the ciphertexts in the header. However, every other device can easily decrypt using one of the keys at its disposal. For example device number 6 can use k14 to decrypt c3 . In e↵ect, changing the encryption scheme to encrypt as in (5.35) revokes device number 3, without impacting any other device. The cost to this is that the ciphertext header now contains log2 n blocks, as opposed to a single block before the device was revoked. More generally, suppose r devices have been compromised and need to be revoked. Let S ✓ {1, . . . , n} be the set of noncompromised devices, so that that S = n r. New content will be encrypted using keys in the tree so that devices in S can decrypt, but all devices outside of S cannot. The set of keys that makes this possible is characterized by the following definition: 201
Definition 5.4. Let T be a complete binary tree with n leaves, where n is a power of two. Let S ✓ {1, . . . , n} be a set of leaves. We say that a set of nodes W ✓ {1, . . . , 2n 1} covers the set S if every leaf in S is a descendant of some node in W , and leaves outside of S are not. We use cover(S) to denote the smallest set of nodes that covers S. Fig. 5.6 gives an example of a cover of the set of leaves {1, 2, 4, 5, 6}. The figure captures a settings where devices number 3, 7, and 8 are revoked. It should be clear that if we use keys in cover(S) to encrypt a movie m, then devices in S can decrypt, but devices outside of S cannot. In particular, we encrypt m as follows: 9 8 R K k > > > > = < for u 2 cover(S) : cu R E(ku , k) . (5.35) cm := R 0 E (k, m) c > > > > ; : output ({cu }u2cover(S) , c) The more devices are revoked, the larger the header of cm becomes. The following theorem shows how big the header gets in the worst case. The proof is an induction argument that also suggests an efficient recursive algorithm to compute an optimal cover.
Theorem 5.8. Let T be a complete binary tree with n leaves, where n is a power of two. For every 1 r n, and every set S of n r leaves, we have cover(S) r · log2 (n/r) Proof. We prove the theorem by induction on log2 n. For n = 1 the theorem is trivial. Now, assume the theorem holds for a tree with n/2 leaves, and let us prove it for a tree T with n leaves. The tree T is made up of a root node, and two disjoint subtrees, T1 and T2 , each with n/2 leaves. Let us split the set S ✓ {1, . . . , n} in two: S = S1 [ S2 , where S1 is contained in {1, . . . , n/2}, and S2 is contained in {n/2 + 1, . . . , n}. That is, S1 are the elements of S that are leaves in T1 , and S2 are the elements of S that are leaves in T2 . Let r1 := (n/2) S1  and r2 := (n/2) S2 . Then clearly r = r1 + r2 . First, suppose both r1 and r2 are greater than zero. By the induction hypothesis, we know that for i = 1, 2 we have cover(Si ) ri log2 (n/2ri ). Therefore, cover(S) = cover(S1 ) + cover(S2 ) r1 log2 (n/2r1 ) + r2 log2 (n/2r2 ) = r log2 (n/r) + r log2 r
r1 log2 (2r1 )
r2 log2 (2r2 ) r log2 (n/r),
which is what we had to prove in the induction step. The last inequality follows from a simple fact about logarithms, namely that for all numbers r1 1 and r2 1, we have (r1 + r2 ) log2 (r1 + r2 ) r1 log2 (2r1 ) + r2 log2 (2r2 ). Second, if r1 = 0 then r2 = r
1, and the induction step follows from:
cover(S) = 1 + cover(S2 ) 1 + r log2 (n/2r) = 1 + r log2 (n/r)
r r log2 (n/r),
as required. The case r2 = 0 follows similarly. This completes the induction step, and the proof. 2 Theorem 5.8 shows that r devices can be revoked at the cost of increasing the ciphertext header size to O(r log n) blocks. For moderate values of r this is not too big. Nevertheless, this general 202
k15 k13
k14
k9 k1
k10 k2
k3
k11 k4
k5
k12 k6
k7
k8
Figure 5.6: The three shaded nodes are the minimal cover for {1, 2, 4, 5, 6}. approach can be improved [82, 51, 48]. The best system using this approach embeds O(log n) keys in every device, same as here, but the header size is only O(r) blocks. The AACS system uses the subsettree di↵erence method [82], which has a worst case header of size 2r 1 blocks, but stores 2 1 2 log n keys per device. While AACS is a far better designed than CSS, it too has been attacked. In particular, the process of a revoking an AACS key is fairly involved and can take several months. For a while, it seemed that hackers could extract new device keys from unrevoked players faster than the industry could revoke them.
5.7
Notes
Citations to the literature to be added.
5.8
Exercises
5.1 (Double encryption). Let E = (E, D) be a cipher. Consider the cipher E2 = (E2 , D2 ), where E2 (k, m) = E(k, E(k, m)). One would expect that if encrypting a message once with E is secure then encrypting it twice as in E2 should be no less secure. However, that is not always true. (a) Show that there is a semantically secure cipher E such that E2 is not semantically secure. (b) Prove that for every CPA secure ciphers E, the cipher E2 is also CPA secure. That is, show that for every CPA adversary A attacking E2 there is a CPA adversary B attacking E with about the same advantage and running time. 5.2 (Multikey CPA security). Generalize the definition of CPA security to the multikey setting, analogous to Definition 5.1. In this attack game, the adversary gets to obtain encryptions of many messages under many keys. The game begins with the adversary outputting a number Q indicating the number of keys it wants to attack. The challenger chooses Q random keys. In every subsequent encryption query, the adversary submits a pair of messages and specifies under which of the Q keys it wants to encrypt; the challenger responds with an encryption of either the first or second message under the specified key (depending on whether the challenger is running Experiment 0 or 1). Flesh out all the details of this attack game, and prove, using a hybrid argument, that (singlekey) CPA security implies multikey CPA security. You should show that security degrades linearly in Q. That is, the advantage of any adversary A in breaking the multikey 203
CPA security of a scheme is at most Q · ✏, where ✏ is the advantage of an adversary B (which is an elementary wrapper around A) in attacking the scheme’s (singlekey) CPA security. 5.3 (An alternate definition of CPA security). This exercise develops an alternative characterization of CPA security for a cipher E = (E, D), defined over (K, M, C). As usual, we need to define an attack game between an adversary A and a challenger. Initially, the challenger generates b
R
{0, 1}, k
R
K.
Then A makes a series of queries to the challenger. There are two types of queries: Encryption: In an encryption query, A submits a message m 2 M to the challenger, who responds with a ciphertext c R E(k, m). The adversary may make any (polybounded) number of encryption queries. Test: In a test query, A submits a pair of messages m0 , m1 2 M to the challenger, who responds with a ciphertext c R E(k, mb ). The adversary is allowed to make only a single test query (with any number of encryption queries before and after the test query). At the end of the game, A outputs a bit ˆb 2 {0, 1}.
As usual, we define A’s advantage in the above attack game to be Pr[ˆb = b] E is AltCPA secure if this advantage is negligible for all efficient adversaries.
1/2. We say that
Show that E is CPA secure if and only if E is AltCPA secure. 5.4 (Hybrid CPA construction). Let (E0 , D0 ) be a semantically secure cipher defined over (K0 , M, C0 ), and let (E1 , D1 ) be a CPA secure cipher defined over (K, K0 , C1 ). (a) Define the following hybrid cipher (E, D) as: E(k, m) := k0 D k, (c1 , c0 ) := k0
R
K 0 , c1
R
E1 (k, k0 ), c0
D1 (k, c1 ), m
R
E0 (k0 , m), output (c1 , c0 )
D0 (k0 , c0 ), output m
Here c1 is called the ciphertext header, and c0 is called the ciphertext body. Prove that (E, D) is CPA secure. (b) Suppose m is some large copyrighted content. A nice feature of (E, D) is that the content owner can make the long ciphertext body c0 public for anyone to download at their leisure. Suppose both Alice and Bob take the time to download c0 . When later Alice, who has key ka , pays for access to the content, the content owner can quickly grant her access by sending her the short ciphertext header ca R E1 (ka , k0 ). Similarly, when Bob, who has key kb , pays for access, the content owner grants him access by sending him the short header cb R E1 (kb , k0 ). Now, an eavesdropper gets to see E 0 (ka , kb ), m := (ca , cb , c0 ) Generalize your proof from part (a) to show that this cipher is also CPA secure. 5.5 (A simple proof of randomized counter mode security). As mentioned in Remark 5.3, we can view randomized counter mode as a special case of the generic hybrid construction in Section 5.4.1. To this end, let F be a PRF defined over (K, X , Y), where X = {0, . . . , N 1} and 204
Y = {0, 1}n , where N is superpoly. For polybounded ` 1, consider the PRF F 0 defined over (K, X , Y ` ) as follows: ⇣ ⌘ F 0 (k, x) := F (k, x), F (k, x + 1 mod N ), . . . , F (k, x + ` 1 mod N ) . (a) Show that F 0 is a weakly secure PRF, as in Definition 4.3.
(b) Using part (a) and Remark 5.2, give a short proof that randomized counter mode is CPA secure. 5.6 (CPA security from a block cipher). Let E = (E, D) be a block cipher defined over (K, M ⇥ R). Consider the cipher E 0 = (E 0 , D0 ), where E 0 (k, m) := r
R
R, c
D0 (k, c) := (m, r0 )
R
E k, (m, r) , output c
D(k, c), output m
This cipher is defined over (K, M, M ⇥ R). Show that if E is a secure block cipher, and 1/R is negligible, then E 0 is CPA secure. 5.7 (pseudorandom ciphertext security). In Exercise 3.4, we developed a notion of security called pseudorandom ciphertext security. This notion naturally extends to multiple ciphertexts. For a cipher E = (E, D) defined over (K, M, C), we define two experiments: in Experiment 0 the challenger first picks a random key k R K and then the adversary submits a sequence of queries, where the ith query is a message mi 2 M, to which the challenger responds with E(k, mi ). Experiment 1 is the same as Experiment 0 except that the challenger responds to the adversary’s queries with random, independent elements of C. We say that E is psuedorandom multiciphertext secure if no efficient adversary can distinguish between these two experiments with a nonnegligible advantage. (a) Consider the countermode construction in Section 5.4.2, based on a PRF F defined over (K, X , Y), but with a fixedlength plaintext space Y ` and a corresponding fixedlength ciphertext space X ⇥ Y ` . Under the assumptions that F is a secure PRF, X  is superpoly, and ` is polybounded, show that this cipher is psuedorandom multiciphertext secure. (b) Consider the CBC construction Section 5.4.3, based on a block cipher E = (E, D) defined over (K, X ), but with a fixedlength plaintext space X ` and corresponding fixedlength ciphertext space X `+1 . Under the assumptions that E is a secure block cipher, X  is superpoly, and ` is polybounded, show that this cipher is psuedorandom multiciphertext secure. (c) Show that a psuedorandom multiciphertext secure cipher is also CPA secure. (d) Give an example of a CPA secure cipher that is not psuedorandom multiciphertext secure. 5.8 (Deterministic CPA and SIV). We have seen that any cipher that is CPA secure must be probabilistic, since for a deterministic cipher, an adversary can always see if the same message is encrypted twice. We may define a relaxed notion of CPA security that says that this is the only thing the adversary can see. This is easily done by placing the following restriction on the adversary in Attack Game 5.2: for all indices i, j, we insist that mi0 = mj0 if and only if mi1 = mj1 . We say that a cipher is deterministic CPA secure if every efficient adversary has negligible advantage 205
in this restricted CPA attack game. In this exercise, we develop a general approach for building deterministic ciphers that are deterministic CPA secure. Let E = (E, D) be a CPAsecure cipher defined over (K, M, C). We let E(k, m; r) denote running algorithm E(k, m) with randomness r R R (for example, if E implements counter mode or CBC encryption then r is the random IV used by algorithm E). Let F be a secure PRF defined over (K0 , M, R). Define the deterministic cipher E 0 = (E 0 , D0 ), defined over (K ⇥ K0 , M, C) as follows: E 0 (k, k 0 ), m D0 (k, k 0 ), c
:= E(k, m; F (k 0 , m)), := D(k, c) .
Show that E 0 is deterministic CPA secure. This construction is known as the Synthetic IV (or SIV) construction. 5.9 (Generic noncebased encryption and nonce reuse resilience). In the previous exercise, we saw how we could generically convert a probabilistic CPAsecure cipher into a deterministic cipher that satisfies a somewhat weaker notion of security called deterministic CPA security. (a) Show how to modify that construction so that we can convert any CPAsecure probabilistic cipher into a noncebased CPAsecure cipher. (b) Show how to combine the two approaches to get a cipher that is noncebased CPA secure, but also satisfies the definition of deterministic CPA security if we drop the uniqueness requirement on nonces. Discussion: This is an instance of a more general security property called nonce reuse resilience: the scheme provides full security if nonces are unique, and even if they are not, a weaker and still useful security guarantee is provided. 5.10 (Ciphertext expansion vs. security). Let E = (E, D) be an encryption scheme messages and ciphertexts are bit strings. (a) Suppose that for all keys and all messages m, the encryption of m is the exact same length as m. Show that (E, D) cannot be semantically secure under a chosen plaintext attack. (b) Suppose that for all keys and all messages m, the encryption of m is exactly ` bits longer than the length of m. Show an attacker that can win the CPA security game using ⇡ 2`/2 queries and advantage ⇡ 1/2. You may assume the message space contains more than ⇡ 2`/2 messages. 5.11 (Repeating ciphertexts). Let E = (E, D) be a cipher defined over (K, M, C). Assume that there are at least two messages in M, that all messages have the same length, and that we can efficiently generate messages in M uniformly at random. Show that if E is CPA secure, then it is infeasible for an adversary to make an encryptor generate the same ciphertext twice. The precise attack game is as follows. The challenger chooses k 2 K at random and the adversary make a series of queries; the ith query is a message mi , to which the challenger’ responds with ci R E(k, mi ). The adversary wins the game if any two ci ’s are the same. Show that if E is CPA secure, then every efficient adversary wins this game with negligible probability. In particular, show that the advantage of any adversary A in winning the repeatedciphertext attack game is at most 2✏, where ✏ is the advantage of an adversary B (which is an elementary wrapper around A) that breaks the scheme’s CPA security. 206
5.12 (Predictable IVs). Let us see why in CBC mode an unpredictable IV is necessary for CPA security. Suppose a defective implementation of CBC encrypts a sequence of messages by always using the last ciphertext block of the ith message as the IV for the (i + 1)st message. The TLS 1.0 protocol, used to protect Web traffic, implements CBC encryption this way. Construct an efficient adversary that wins the CPA game against this implementation with advantage close to 1. We note that the Webbased BEAST attack [35] exploits this defect to completely break CBC encryption in TLS 1.0. 5.13 (CBC encryption with small blocks is insecure). Suppose the block cipher used for CBC encryption has a block size of n bits. Construct an attacker that wins the CPA game against CBC that makes ⇡ 2n/2 queries to its challenger and gains an advantage ⇡ 1/2. Your answer explains why CBC cannot be used with a block cipher that has a small block size (e.g. n = 64 bits). This is one reason why AES has a block size of 128 bits. Discussion: This attack was used to show that 3DES is no longer secure for Internet use, due to its 64bit block size [11]. 5.14 (An insecure noncebased CBC mode). Consider the noncebased CBC scheme E 0 described in Section 5.5.3. Suppose that the nonce space N is equal to block space X of the underlying block cipher E = (E, D), and the PRF F is just the encryption algorithm E. If the two keys k and k 0 in the construction are chosen independently, the scheme is secure. Your task is to show that if only one key k is chosen, and other key k 0 is just set to k, then the scheme is insecure. 5.15 (Output feedback mode). Suppose F is a PRF defined over (K, X ), and ` bounded.
1 is poly
(a) Consider the following PRG G : K ! X ` . Let x0 be an arbitrary, fixed element of X . For k 2 K, let G(k) := (x1 , . . . , x` ), where xi := F (k, xi 1 ) for i = 1, . . . , `. Show that G is a secure PRG, assuming F is a secure PRF and that X  is superpoly. (b) Next, assume that X = {0, 1}n . We define a cipher E = (E, D), defined over (K, X ` , X `+1 ), as follows. Given a key k 2 K and a message (m1 , . . . , m` ) 2 X ` , the encryption algorithm E generates the ciphertext (c0 , c1 , . . . , c` ) 2 X `+1 as follows: it chooses x0 2 X at random, and sets c0 = x0 ; it then computes xi = F (k, xi 1 ) and ci = mi xi for i = 1, . . . , `. Describe the corresponding decryption algorithm D, and show that E is CPA secure, assuming F is a secure PRF and that X  is superpoly. Note: This construction is called output feedback mode (or OFB).
5.16 (CBC ciphertext stealing). One problem with CBC encryption is that messages need to be padded to a multiple of the block length and sometimes a dummy block needs to be added. The following figure describes a variant of CBC that eliminates the need to pad:
207
The method pads the last block with zeros if needed (a dummy block is never added), but the output ciphertext contains only the shaded parts of C1 , C2 , C3 , C4 . Note that, ignoring the IV, the ciphertext is the same length as the plaintext. This technique is called ciphertext stealing. (a) Explain how decryption works. (b) Can this method be used if the plaintext contains only one block? 5.17 (Single ciphertext block corruption in CBC mode). Let c be an ` block CBCencrypted ciphertext, for some ` > 3. Suppose that exactly one block of c is corrupted, and the result is decrypted using the CBC decryption algorithm. How many blocks of the decrypted plaintext are corrupted? 5.18 (The malleability of CBC mode). Let c be the CBC encryption of some message m 2 X ` , 2 X . Show how to modify the ciphertext c to where X := {0, 1}n . You do not know m. Let , and m0 [i] = m[i] for obtain a new ciphertext c0 that decrypts to m0 , where m0 [0] = m[0] i = 1, . . . , ` 1. That is, by modifying c appropriately, you can flip bits of your choice in the first block of the decryption of c, without a↵ecting any the other blocks. 5.19 (Online ciphers). In practice there is a strong desire to encrypt one block of plaintext at a time, outputting the corresponding block of ciphertext right away. This lets the system transmit ciphertext blocks as soon as they are ready without having to wait until the entire message is processed by the encryption algorithm. (a) Define a CPAlike security game that captures this method of encryption. Instead of forcing the adversary to submit a complete pair of messages in every encryption query, the adversary should be allowed to issue a query indicating the beginning of a message, then repeatedly issue more queries containing message blocks, and finally issue a query indicating the end of a message. Responses to these queries will include all ciphertext blocks that can be computed given the information given. (b) Show that randomized CBC encryption is not CPA secure in this model. (c) Show that randomized counter mode is online CPA secure. 5.20 (Redundant bits do not harm CPA security). Let E = (E, D) be a CPAsecure cipher defined over (K, M, C). Show that appending to a ciphertext additional data that is computed from the ciphertext does not damage CPA security. Specifically, let g : C ! Y be some efficiently computable function. Show that the following modified cipher E 0 = (E 0 , D0 ) is CPAsecure: E(k, m), t E 0 (k, m) := c 0 := D k, (c, t) D(k, c)
g(c), output (c, t)
208
Chapter 6
Message integrity In previous chapters we focused on security against an eavesdropping adversary. The adversary had the ability to eavesdrop on transmitted messages, but could not change messages enroute. We showed that chosen plaintext security is the natural security property needed to defend against such attacks. In this chapter we turn our attention to active adversaries. We start with the basic question of message integrity: Bob receives a message m from Alice and wants to convince himself that the message was not modified enroute. We will design a mechanism that lets Alice compute a short message integrity tag t for the message m and send the pair (m, t) to Bob, as shown in Fig. 6.1. Upon receipt, Bob checks the tag t and rejects the message if the tag fails to verify. If the tag verifies then Bob is assured that the message was not modified in transmission. We emphasize that in this chapter the message itself need not be secret. Unlike previous chapters, our goal here is not to conceal the message. Instead, we only focus on message integrity. In Chapter 9 we will discuss the more general question of simultaneously providing message secrecy and message integrity. There are many applications where message integrity is needed, but message secrecy is not. We give two examples. Example 6.1. Consider the problem of delivering financial news or stock quotes over the Internet. Although the news items themselves are public information, it is vital that no third party modify the data on its way to the user. Here message secrecy is irrelevant, but message integrity is critical. Our constructions will ensure that if user Bob rejects all messages with an invalid message integrity tag then an attacker cannot inject modified content that will look legitimate. One caveat is that an attacker can still change the order in which news reports reach Bob. For example, Bob might see report number 2 before seeing report number 1. In some settings this may cause the user to take an incorrect action. To defend against this, the news service may wish to include a sequence number with each report so that the user’s machine can bu↵er reports and ensure that the user always sees news items in the correct order. 2 In this chapter we are only concerned with attacks that attempt to modify data. We do not consider Denial of Service (DoS) attacks, where the attacker delays or prevents news items from reaching the user. DoS attacks are often handled by ensuring that the network contains redundant paths from the sender to the receiver so that an attacker cannot block all paths. We will not discuss these issues here. Example 6.2. Consider an application program — such as a word processor or mail client — 209
m
m
t
m
Alice
Bob
Generate tag t
Verify messagetag pair (m, t)
t
?
S(k, m)
V (k, m, t) = accept
Figure 6.1: Short message integrity tag added to messages stored on disk. Although the application code is not secret (it might even be in the public domain), its integrity is important. Before running the program the user wants to ensure that a virus did not modify the code stored on disk. To do so, when the program is first installed, the user computes a message integrity tag for the code and stores the tag on disk alongside the program. Then, every time, before starting the application the user can validate this message integrity tag. If the tag is valid, the user is assured that the code has not been modified since the tag was initially generated. Clearly a virus can overwrite both the application code and the integrity tag. Nevertheless, our constructions will ensure that no virus can fool the user into running unauthenticated code. As in our first example, the attacker can swap two authenticated programs — when the user starts application A he will instead be running application B. If both applications have a valid tag the system will not detect the swap. The standard defense against this is to include the program name in the executable file. That way, when an application is started the system can display to the user an authenticated application name. 2 The question, then, is how to design a secure message integrity mechanism. We first argue the following basic principle: Providing message integrity between two communicating parties requires that the sending party has a secret key unknown to the adversary. Without a secret key, ensuring message integrity is not possible: the adversary has enough information to compute tags for arbitrary messages of its choice — it knows how the message integrity algorithm works and needs no other information to compute tags. For this reason all cryptographic message integrity mechanisms require a secret key unknown to the adversary. In this chapter, we will assume that both sender and receiver will share the secret key; later in the book, this assumption will be relaxed. We note that communication protocols not designed for security often use keyless integrity mechanisms. For example, the Ethernet protocol uses CRC32 as its message integrity algorithm. This algorithm, which is publicly available, outputs 32bit tags embedded in every Ethernet frame. The TCP protocol uses a keyless 16bit checksum which is embedded in every packet. We emphasize that these keyless integrity mechanisms are designed to detect random transmission errors, not malicious errors. The argument in the previous paragraph shows that an adversary can easily defeat these mechanisms and generate legitimatelooking traffic. For example, in the case of Ethernet, the adversary knows exactly how the CRC32 algorithm works and this lets him compute valid tags for arbitrary messages. He can then tamper with Ethernet traffic without being detected.
210
6.1
Definition of a message authentication code
We begin by defining what is a message integrity system based on a shared secret key between the sender and receiver. For historical reasons such systems are called Message Authentication Codes or MACs for short. Definition 6.1. A MAC system I = (S, V ) is a pair of efficient algorithms, S and V , where S is called a signing algorithm and V is called a verification algorithm. Algorithm S is used to generate tags and algorithm V is used to verify tags. • S is a probabilistic algorithm that is invoked as t and the output t is called a tag.
R
S(k, m), where k is a key, m is a message,
• V is a deterministic algorithm that is invoked as r V (k, m, t), where k is a key, m is a message, t is a tag, and the output r us either accept or reject. • We require that tags generated by S are always accepted by V ; that is, the MAC must satisfy the following correctness property: for all keys k and all messages m, Pr[V (k, m, S(k, m) ) = accept] = 1. As usual, we say that keys lie in some finite key space K, messages lie in a finite message space M, and tags lie in some finite tag space T . We say that I = (S, V ) is defined over (K, M, T ). Fig. 6.1 illustrates how algorithms S and V are used for protecting network communications between two parties. Whenever algorithm V outputs accept for some messagetag pair (m, t), we say that t is a valid tag for m under key k, or that (m, t) is a valid pair under k. Naturally, we want MAC systems where tags are as short as possible so that the overhead of transmitting the tag is minimal. We will explore a variety of MAC systems. The simplest type of system is one in which the signing algorithm S is deterministic, and the verification algorithm is defined as ( accept if S(k, m) = t, V (k, m, t) = reject otherwise. We shall call such a MAC system a deterministic MAC system. One property of a deterministic MAC system is that it has unique tags: for a given key k, and a given message m, there is a unique valid tag for m under k. Not all MAC systems we explore will have such a simple design: some have a randomized signing algorithm, so that for a given key k and message m, the output of S(k, m) may be one of many possible valid tags, and the verification algorithm works some other way. As we shall see, such randomized MAC systems are not necessary to achieve security, but they can yield better efficiency/security tradeo↵s. Secure MACs. Next, we turn to describing what it means for a MAC to be secure. To construct MACs that remain secure in a variety of applications we will insist on security in a very hostile environment. Since most realworld systems that use MACs operate in less hostile settings, our conservative security definitions will imply security for all these systems. We first intuitively explain the definition and then motivate why this conservative definition makes sense. Suppose an adversary is attacking a MAC system I = (S, V ). Let k be some 211
Adversary A
MAC Challenger k
R
K
mi ti
S(k, mi ) (m, t)
Figure 6.2: MAC attack game (Attack Game 6.1) randomly chosen MAC key, which is unknown to the attacker. We allow the attacker to request tags t := S(k, m) for arbitrary messages m of its choice. This attack, called a chosen message attack, enables the attacker to collect millions of valid messagetag pairs. Clearly we are giving the attacker considerable power — it is hard to imagine that a user would be foolish enough to sign arbitrary messages supplied by an attacker. Nevertheless, we will see that chosen message attacks come up in real world settings. We refer to messagetag pairs (m, t) that the adversary obtains using the chosen message attack as signed pairs. Using the chosen message attack we ask the attacker to come up with an existential MAC forgery. That is, the attacker need only come up with some new valid messagetag pair (m, t). By “new”, we mean a messagetag pair that is di↵erent from all of the signed pairs. The attacker is free to choose m arbitrarily; indeed, m need not have any special format or meaning and can be complete gibberish. We say that a MAC system is secure if even an adversary who can mount a chosen message attack cannot create an existential forgery. This definition gives the adversary more power than it typically has in the real world and yet we ask it to do something that will normally be harmless; forging the MAC for a meaningless message seems to be of little use. Nevertheless, as we will see, this conservative definition is very natural and enables us to use MACs for lots of di↵erent applications. More precisely, we define secure MACs using an attack game between a challenger and an adversary A. The game is described below and in Fig. 6.2. Attack Game 6.1 (MAC security). For a given MAC system I = (S, V ), defined over (K, M, T ), and a given adversary A, the attack game runs as follows: • The challenger picks a random k
R
K.
• A queries the challenger several times. For i = 1, 2, . . . , the ith signing query is a message mi 2 M. Given mi , the challenger computes a tag ti R S(k, mi ), and then gives ti to A. • Eventually A outputs a candidate forgery pair (m, t) 2 M ⇥ T that is not among the signed pairs, i.e., (m, t) 62 (m1 , t1 ), (m2 , t2 ), . . . . We say that A wins the above game if (m, t) is a valid pair under k (i.e., V (k, m, t) = accept). We define A’s advantage with respect to I, denoted MACadv[A, I], as the probability that A wins 212
the game. Finally, we say that A is a Qquery MAC adversary if A issues at most Q signing queries. 2 Definition 6.2. We say that a MAC system I is secure if for all efficient adversaries A, the value MACadv[A, I] is negligible. In case the adversary wins Attack Game 6.1, the pair (m, t) it sends the challenger is called an existential forgery. MAC systems that satisfy Definition 6.2 are said to be existentially unforgeable under a chosen message attack. In the case of a deterministic MAC system, the only way for A to win Attack Game 6.1 is to produce a valid messagetag pair (m, t) for some new message m 2 / {m1 , m2 , . . .}. Indeed, security in this case just means that S is unpredictable, in the sense described in Section 4.1.1; that is, given / {m1 , m2 , . . .}. S(k, m1 ), S(k, m2 ), . . . , it is hard to predict S(k, m) for any m 2 In the case of a randomized MAC system, our security definition captures a stronger property. There may be many valid tags for a given message. Let m be some message and suppose the adversary requests one or more valid tags t1 , t2 , . . . for m. Can the adversary produce a new valid / {t1 , t2 , . . .}). Our definition says that a valid pair (m, t0 ), tag t0 for m? (i.e. a tag satisfying t0 2 0 where t is new, is a valid existential forgery. Therefore, for a MAC to be secure it must be difficult for an adversary to produce a new valid tag t0 for a previously signed message m. This may seem like an odd thing to require of a MAC. If the adversary already has valid tags for m, why should we care if it can produce another one? As we will see in Chapter 9, our security definition, which prevents the adversary from producing new tags on signed messages, is necessary for the applications we have in mind. Going back to the examples in the introduction, observe that existential unforgeability implies that an attacker cannot create a fake news report with a valid tag. Similarly, the attacker cannot tamper with a program on disk without invalidating the tag for the program. Note, however, that when using MACs to protect application code, users must provide their secret MAC key every time they want to run the application. This will quickly annoy most users. In Chapter 8 we will discuss a keyless method to protect public application code. To exercise the definition of secure MACs let us first see a few consequences of it. Let I = (S, V ) be a MAC defined over (K, M, T ), and let k be a random key in K. Example 6.3. Suppose m1 and m2 are almost identical messages. Say m1 is a money transfer order for $100 and m2 is a transfer order for $101. Clearly, an adversary who intercepts a valid tag for m1 should not be able to deduce from it a valid tag for m2 . A MAC system that satisfies Definition 6.2 ensures this. To see why, suppose an adversary A can forge the tag for m2 given the tag for m1 . Then A can win Attack Game 6.1: it uses the chosen message attack to request a tag for m1 , deduces a forged tag t2 for m2 , and outputs (m2 , t2 ) as a valid existential forgery. Clearly A wins Attack Game 6.1. Hence, existential unforgeability captures the fact that a tag for one message m1 gives no useful information for producing a tag for another message m2 , even when m2 is almost identical to m1 . 2 Example 6.4. Our definition of secure MACs gives the adversary the ability to obtain the tag for arbitrary messages. This may seem like giving the adversary too much power. In practice, however, there are many scenarios where chosen message attacks are feasible. The reason is that the MAC signer often does not know the source of the data being signed. For example, consider a backup system that dumps the contents of disk to backup tapes. Since backup integrity is important, the 213
system computes an integrity tag on every disk block that it writes to tape. The tag is stored on tape along with the data block. Now, suppose an attacker writes data to a low security part of disk. The attacker’s data will be backed up and the system will compute a tag over it. By examining the resulting backup tape the attacker obtains a tag on his chosen message. If the MAC system is secure against a chosen message attack then this does not help the attacker break the system. 2 Remark 6.1. Just as we did for other security primitives, one can generalize the notion of a secure MAC to the multikey setting, and prove that a secure MAC is also secure in the multikey setting. See Exercise 6.3. 2
6.1.1
Mathematical details
As usual, we give a more mathematically precise definition of a MAC, using the terminology defined in Section 2.4. This section may be safely skipped on first reading. Definition 6.3 (MAC). A MAC system is a pair of efficient algorithms, S and V , along with three families of spaces with system parameterization P : K = {K As usual, that
2Z
1
,⇤ } ,⇤ ,
M = {M
,⇤ } ,⇤ ,
and
T = {T
,⇤ } ,⇤ ,
is a security parameter and ⇤ 2 Supp(P ( )) is a domain parameter. We require
1. K, M, and T are efficiently recognizable. 2. K is efficiently sampleable. 2Z
1,
4. Algorithm V is an efficient deterministic algorithm that on input , ⇤, k, m, t, where 2 Z ⇤ 2 Supp(P ( )), k 2 K ,⇤ , m 2 M ,⇤ , and t 2 T ,⇤ , outputs either accept or reject.
1,
3. Algorithm S is an efficient probabilistic algorithm that on input , ⇤, k, m, where ⇤ 2 Supp(P ( )), k 2 K ,⇤ , and m 2 M ,⇤ , outputs an element of T ,⇤ .
In defining security, we parameterize Attack Game 6.1 by the security parameter , which is given to both the adversary and the challenger. The advantage MACadv[A, I] is then a function of . Definition 6.2 should be read as saying that MACadv[A, I]( ) is a negligible function.
6.2
MAC verification queries do not help the attacker
In our definition of secure MACs (Attack Game 6.1) the adversary has no way of testing whether a given messagetag pair is valid. In fact, the adversary cannot even tell if it wins the game, since only the challenger has the secret key needed to run the verification algorithm. In real life, an attacker capable of mounting a chosen message attack can probably also test whether a given messagetag pair is valid. For example, the attacker could build a packet containing the messagetag pair in question and send this packet to the victim’s machine. Then, by examining the machine’s behavior the attacker can tell whether the packet was accepted or dropped, indicating whether the tag was valid or not. Consequently, it makes sense to extend Attack Game 6.1 by giving the adversary the extra power to verify messagetag pairs. Of course, we continue to allow the adversary to request tags for arbitrary messages of his choice. 214
Attack Game 6.2 (MAC security with verification queries). For a given MAC system I = (S, V ), defined over (K, M, T ), and a given adversary A, the attack game runs as follows: • The challenger picks a random k
R
K.
• A queries the challenger several times. Each query can be one of two types:
– Signing query: for i = 1, 2, . . . , the ith signing query consists of a message mi 2 M. The challenger computes a tag ti R S(k, mi ), and gives ti to A. – Verification query: for j = 1, 2, . . . , the jth verification query consists of a messagetag pair (m ˆ j , tˆj ) 2 M ⇥ T that is not among the previously signed pairs, i.e., (m ˆ j , tˆj ) 62 (m1 , t1 ), (m2 , t2 ), . . . . The challenger responds to A with V (k, m ˆ j , tˆj ).
We say that A wins the above game if the challenger ever responds to a verification query with accept. We define A’s advantage with respect to I, denoted MACvq adv[A, I], as the probability that A wins the game. 2 The two definitions are equivalent. Attack Game 6.2 is essentially the same as the original Attack Game 6.1, except that A can issue MAC verification queries. We prove that this extra power does not help the adversary. Theorem 6.1. If I is a secure MAC system, then it is also secure in the presence of verification queries. In particular, for every MAC adversary A that attacks I as in Attack Game 6.2, and which makes at most Qv verification queries and at most Qs signing queries, there exists a Qs query MAC adversary B that attacks I as in Attack Game 6.1, where B is an elementary wrapper around A, such that MACvq adv[A, I] MACadv[B, I] · Qv .
Proof idea. Let A be a MAC adversary that attacks I as in Attack Game 6.2, and which makes at most Qv verification queries and at most Qs signing queries. From adversary A, we build an adversary B that attacks I as in Attack Game 6.1 and makes at most Qs signing queries. Adversary B can easily answer A’s signing queries by forwarding them to B’s challenger and relaying the resulting tags back to A. The question is how to respond to A’s verification queries. Note that A by definition, A only submits verification queries on message pairs that are not among the previously signed pairs. So B adopts a simple strategy: it responds with reject to all verification queries from A. If B answers incorrectly, it has a forgery which would let it win Attack Game 6.1. Unfortunately, B does not know which of these verification queries is a forgery, so it simply guesses, choosing one at random. Since A makes at most Qv verification queries, B will guess correctly with probability at least 1/Qv . This is the source of the Qv factor in the error term. 2 Proof. In more detail, adversary B plays the role of challenger to A in Attack Game 6.2, while at the same time, it plays the role of adversary in Attack Game 6.1, interacting with the MAC challenger in that game. The logic is as follows:
215
initialization: ! R {1, . . . , Qv }
upon receiving a signing query mi 2 M from A do: forward mi to the MAC challenger, obtaining the tag ti send ti to A upon receiving a verification query (m ˆ j , tˆj ) 2 M ⇥ T from A do: if j = ! then output (m ˆ j , tˆj ) as a candidate forgery pair and halt else send reject to A
To rigorously justify the construction of adversary B, we analyze the the behavior of A in three closely related games. Game 0. This is the original attack game, as played between the challenger in Attack Game 6.2 and adversary A. Here is the logic of the challenger in this game: initialization: k R K
upon receiving a signing query mi 2 M from A do: ti R S(k, mi ) send ti to A upon receiving a verification query (m ˆ j , tˆj ) 2 M ⇥ T from A do: V (k, m ˆ j , tˆj ) rj (⇤) send rj to A Let W0 be the event that in Game 0, rj = accept for some j. Evidently, Pr[W0 ] = MACvq adv[A, I].
(6.1)
Game 1. This is the same as Game 1, except that the line marked (⇤) above is changed to: send reject to A That is, when responding to a verification query, the challenger always responds to A with reject. We also define W1 to be the event that in Game 1, rj = accept for some j. Even though the challenger does not notify A that W1 occurs, both Games 0 and 1 proceed identically until this event happens, and so events W0 and W1 are really the same; therefore, Pr[W1 ] = Pr[W0 ].
(6.2)
Also note that in Game 1, although the rj values are used to define the winning condition, they are not used for any other purpose, and so do not influence the attack in any way. Game 2. This is the same as Game 1, except that at the beginning of the game, the challenger chooses ! R {1, . . . , Qv }. We define W2 to be the event that in Game 2, r! = accept. Since the choice of ! is independent of the attack itself, we have Pr[W2 ]
Pr[W1 ]/Qv . 216
(6.3)
Evidently, by construction, we have Pr[W2 ] = MACadv[B, I].
(6.4)
The theorem now follows from (6.1)–(6.3). 2 In summary, we showed that Attack Game 6.2, which gives the adversary more power, is equivalent to Attack Game 6.1 used in defining secure MACs. The reduction introduces a factor of Qv in the error term. Throughout the book we will make use of both attack games: • When constructing secure MACs it easier to use Attack Game 6.1 which restricts the adversary to signing queries only. This makes it easier to prove security since we only have to worry about one type of query. We will use this attack game throughout the chapter. • When using secure MACs to build higher level systems (such as authenticated encryption) it is more convenient to assume that the MAC is secure with respect to the stronger adversary described in Attack Game 6.2. We also point out that if we had used a weaker notion of security, in which the adversary only wins by presenting a valid tag on a new message (rather than new valid messagetag pair), then the analogs of Attack Game 6.1 and Attack Game 6.2 are not equivalent (see Exercise 6.7).
6.3
Constructing MACs from PRFs
We now turn to constructing secure MACs using the tools at our disposal. In previous chapters we used pseudo random functions (PRFs) to build various encryption systems. We gave examples of practical PRFs such as AES (while AES is a block cipher it can be viewed as a PRF thanks to the PRF switching lemma, Theorem 4.4). Here we show that any secure PRF can be directly used to build a secure MAC. Recall that a PRF is an algorithm F that takes two inputs, a key k and an input data block x, and outputs a value y := F (k, x). As usual, we say that F is defined over (K, X , Y), where keys are in K, inputs are in X , and outputs are in Y. For a PRF F we define the deterministic MAC system I = (S, V ) derived from F as: S(k, m) := F (k, m); ( accept if F (k, m) = t, V (k, m, t) := reject otherwise. As already discussed, any PRF with a large (i.e., superpoly) output space is unpredictable (see Section 4.1.1), and therefore, as discussed in Section 6.1, the above construction yields a secure MAC. For completeness, we state this as a theorem: Theorem 6.2. Let F be a secure PRF defined over (K, X , Y), where Y is superpoly. Then the deterministic MAC system I derived from F is a secure MAC. In particular, for every Qquery MAC adversary A that attacks I as in Attack Game 6.1, there exists a (Q + 1)query PRF adversary B that attacks F as in Attack Game 4.2, where B is an elementary wrapper around A, such that MACadv[A, I] PRFadv[B, F ] + 1/Y
217
Proof idea. Let A be an efficient MAC adversary. We derive an upper bound on MACadv[A, I] by bounding A’s ability to generate forged messagetag pairs. As usual, replacing the underlying secure PRF F with a truly random function f in Funs[X , Y] does not change A’s advantage much. But now that the adversary A is interacting with a truly random function it is faced with a hopeless task: using the chosen message attack it obtains the value of f at a few points of his choice. He then needs to guess the value of f (m) 2 Y at some new point m. But since f is a truly random function, A has no information about f (m), and therefore has little chance of guessing f (m) correctly. 2 Proof. We make this intuition rigorous by letting A interact with two closely related challengers. Game 0. As usual, we begin by reviewing the challenger in the MAC Attack Game 6.1 as it applies to I. We implement the challenger in this game as follows: (⇤) k R K, f F (k, ·) upon receiving the ith signing query mi 2 M (for i = 1, 2, . . .) do: f (mi ) ti send ti to the adversary At the end of the game, the adversary outputs a messagetag pair (m, t). We define W0 to be the event that the condition t = f (m) and m 62 {m1 , m2 , . . .} (6.5) holds in Game 0. Clearly, Pr[W0 ] = MACadv[A, I]. Game 1. We next play the usual “PRF card,” replacing the function F (k, ·) by a truly random function f in Funs[X , Y]. Intuitively, since F is a secure PRF, the adversary A should not notice the di↵erence. Our challenger in Game 1 is the same as in Game 0 except that we change line (*) as follows: (⇤) f
R
Funs[X , Y]
We define W1 to be the event that condition (6.5) holds in Game 1. It should be clear how to design the corresponding PRF adversary B such that: Pr[W1 ]
Pr[W0 ] = PRFadv[B, F ].
Next, we directly bound Pr[W1 ]. The adversary A sees the values of f at various points m1 , m2 , . . . and is then required to guess the value of f at some new point m. But since f is a truly random function, the value f (m) is independent of its value at all other points. Hence, since m 62 {m1 , m2 , . . .}, adversary A will guess f (m) with probability 1/Y. Therefore, Pr[W1 ] 1/Y. Putting it all together, we obtain MACadv[A, I] = Pr[W0 ] Pr[W0 ]
Pr[W1 ] + Pr[W1 ] PRFadv[B, F ] +
1 Y
as required. 2 Concrete tag lengths. The theorem shows that to ensure MACadv[A, I] < 2 128 we need a PRF whose output space Y satisfies Y > 2128 . If the output space Y is {0, 1}n for some n, then the resulting tags must be at least 128 bits long.
218
6.4
Prefixfree PRFs for long messages
In the previous section we saw that any secure PRF is also a secure MAC. However, the concrete examples of PRFs from Chapter 4 only take short inputs and can therefore only be used to provide integrity for very short messages. For example, viewing AES as a PRF gives a MAC for 128bit messages. Clearly, we want to build MACs for much longer messages. All the MAC constructions in this chapter follow the same paradigm: they start from a PRF for short inputs (like AES) and produce a PRF, and therefore a MAC, for much longer inputs. Hence, our goal for the remainder of the chapter is the following: given a secure PRF on short inputs construct a secure PRF on long inputs. We solve this problem in three steps: • First, in this section we construct prefixfree secure PRFs for long inputs. More precisely, given a secure PRF that operates on singleblock (e.g., 128bit) inputs, we construct a prefixfree secure PRF that operates on variablelength sequences of blocks. Recall that a prefixfree secure PRF (Definition 4.5) is only secure in a limited sense: we only require that prefixfree adversaries cannot distinguish the PRF from a random function. A prefixfree PRF adversary issues queries that are nonempty sequences of blocks, and no query can be a proper prefix of another. • Second, in the next few sections we show how to convert prefixfree secure PRFs for long inputs into fully secure PRFs for long inputs. Thus, by the end of these sections we will have several secure PRFs, and therefore secure MACs, that operate on long inputs. • Third, in Section 6.8 we show how to convert a PRF that operates on messages that are strings of blocks into a PRF that operates on strings of bits. Prefixfree PRFs. We begin with two classic constructions for prefixfree secure PRFs. The CBC construction is shown in Fig. 6.3a. The cascade construction is shown in Fig. 6.3b. We show that when the underlying F is a secure PRF, both CBC and cascade are prefixfree secure PRFs.
6.4.1
The CBC prefixfree secure PRF
Let F be a PRF that maps nbit inputs to nbit outputs. In symbols, F is defined over (K, X , X ) where X = {0, 1}n . For any polybounded value `, we build a new PRF, denoted FCBC , that maps messages in X ` to outputs in X . The function FCBC , described in Fig. 6.3a, works as follows: input: k 2 K and m = (a1 , . . . , av ) 2 X ` for some v 2 {0, . . . , `} output: a tag in X t 0n for i 1 to v do: t F (k, ai output t
t)
219
a1
F (k, ·)
a2
a3
a`
L
L
L
F (k, ·)
F (k, ·)
F (k, ·)
···
tag
(a) The CBC construction FCBC (k, m)
k
a1
a2
a3
F
F
F
a`
···
F
tag
(b) The cascade construction F ⇤ (k, m)
Figure 6.3: Two prefixfree secure PRFs FCBC is similar to CBC mode encryption from Fig. 5.4, but with two important di↵erences. First, FCBC does not output any intermediate values along the CBC chain. Second, FCBC uses a fixed IV, namely 0n , where as CBC mode encryption uses a random IV per message. The following theorem shows that FCBC is a prefixfree secure PRF defined over (K, X ` , X ). Theorem 6.3. Let F be a secure PRF defined over (K, X , X ) where X = {0, 1}n and X  = 2n is superpoly. Then for any polybounded value `, we have that FCBC is a prefixfree secure PRF defined over (K, X ` , X ). In particular, for every prefixfree PRF adversary A that attacks FCBC as in Attack Game 4.2, and issues at most Q queries, there exists a PRF adversary B that attacks F as in Attack Game 4.2, where B is an elementary wrapper around A, such that PRFpf adv[A, FCBC ] PRFadv[B, F ] +
(Q`)2 . 2X 
(6.6)
Exercise 6.6 develops an attack on fixedlength FCBC that demonstrates that security degrades quadratically in Q. This shows that the quadratic dependence on Q in (6.6) is necessary. A more difficult proof of security shows that security only degrades linearly in ` (see Section 6.13). In particular, the error term in (6.6) can be reduced to an expression dominated by O(Q2 `/X ) Proof idea. We represent the adversary’s queries in a rooted tree, where edges in the tree are labeled by message blocks (i.e., elements of X ). A query for FCBC (k, m), where m = (a1 , . . . , av ) 2 X v and 1 v `, defines a path in the tree, starting at the root, as follows: a
a
a
a
1 2 3 v p1 ! p2 ! ··· ! pv . root !
220
(6.7)
Thus, two messages m and m0 correspond to paths in the tree which both start at the root; these two paths may share a common initial subpath corresponding to the longest common prefix of m and m0 . With each node p in this tree, we associate a value p 2 X which represents the computed value in the CBC chain. More precisely, we define root := 0n , and for any nonroot node q with parent a p, if the corresponding edge in the tree is p ! q, then q := F (k, p a). With these conventions, we see that if a message m traces out a path as in (6.7), then pv = FCBC (k, m). The crux of the proof is to argue that if F behaves like a random function, then for every a0
a
a0 with pair of distinct edges in the tree, say p ! q and p0 ! q 0 , we have p a 6= p0 overwhelming probability. To prove that there are no collisions of this type, the prefixfreeness restriction is critical, as it guarantees that the adversary never sees p and p0 , and hence a and a0 are independent of these values. Once we have established that there are no collisions of these types, it will follow that all values associated with nonroot nodes are random and independent, and this holds in particular for the values associated with the leaves, which represent the outputs of FCBC seen by the adversary. Therefore, the adversary cannot distinguish FCBC from a random function. 2 Proof. We make this intuition rigorous by letting A interact with three closely related challengers in three games. For j = 0, 1, 2, 3, we let Wj be the event that A outputs 1 at the end of Game j. Game 0. This is Experiment 0 of Attack Game 4.2. Game 1. We next play the usual “PRF card,” replacing the function F (k, ·) by a truly random function f in Funs[X , X ]. Clearly, we have Pr[W1 ]
Pr[W0 ] = PRFadv[B, F ]
(6.8)
for an efficient adversary B. Game 2. We now make a purely conceptual change, implementing the random function f as a “faithful gnome” (as in Section 4.4.2). However, it will be convenient for us to do this is a particular way, using the “query tree” discussed above. To this end, first let B := Q`, which represents an upper bound on how many points at which f will evaluated. Our challenger first prepares random values i
R
X
(i = 1, . . . , B).
These will be the only random values used by our challenger. As the adversary makes queries, our challenger will dynamically build up the query tree. Initially, the tree contains only the root. Whenever the adversary makes a query, the challenger traces out the corresponding path in the existing query tree; at some point, this path will extend beyond the existing query tree, and our challenger adds the necessary nodes and edges so that the query tree grows to include the new path. Our challenger must also compute the values p associated with each node. Initially, root = 0n . a When adding a new edge p ! q to the tree, if this is the ith edge being added (for i = 1, . . . , B), our challenger does the following: q
(⇤)
i a0
if 9 another edge p0 ! q 0 with
p0
a0 = 221
p
a then
q
q0
The idea is that we use the next unused value in our prepared list 1 , . . . , B as the “default” value for q . The line marked (⇤) performs the necessary consistency check, which ensures that our gnome is indeed faithful. Because this change is purely conceptual, we have Pr[W2 ] = Pr[W1 ].
(6.9)
Game 3. Next, we make our gnome forgetful, by removing the consistency check marked (⇤) in the logic in Game 2. To analyze the e↵ect of this change, let Z be the event that in Game 3, for some distinct pair a
a0
of edges p ! q and p0 ! q 0 , we have p0 a0 = p a. Now, the only randomly chosen values in Games 2 and 3 are the random choices of the adversary, Coins, and the list of values 1 , . . . , B . Observe that for any fixed choice of values Coins, 1 , . . . , B , if Z does not occur, then in fact Games 2 and 3 proceed identically. Therefore, we may apply the Di↵erence Lemma (Theorem 4.7), obtaining Pr[W3 ]
Pr[W2 ] Pr[Z]. a
(6.10) a0
We next bound Pr[Z]. Consider two distinct edges p ! q and p0 ! q 0 . We want to bound the probability that p0 a0 = p a, which is equivalent to p0
p
= a0
a.
(6.11)
There are two cases to consider. Case 1: p = p0 . Since the edges are distinct, we must have a0 6= a, and hence (6.11) holds with probability 0. Case 2: p 6= p0 . The requirement that the adversary’s queries are prefix free implies that in Game 3, the adversary never sees — or learns anything about — the values p and p0 . One of p or p0 could be the root, but not both. It follows that the value p p0 is uniformly distributed over X and is independent of a a0 . From this, it follows that (6.11) holds with probability 1/X . By the union bound, it follows that Pr[Z]
B2 . 2X 
(6.12)
Combining (6.8), (6.9), (6.10), and (6.12), we obtain PRFpf adv[A, FCBC ] = Pr[W3 ]
Pr[W0 ] PRFadv[B, F ] +
B2 . 2X 
(6.13)
Moreover, Game 3 corresponds exactly to Experiment 1 of Attack Game 4.2, from which the theorem follows. 2
6.4.2
The cascade prefixfree secure PRF
Let F be a PRF that takes keys in K and produces outputs in K. In symbols, F is defined over (K, X , K). For any polybounded value `, we build a new PRF F ⇤ , called the cascade of F , that maps messages in X ` to outputs in K. The function F ⇤ , illustrated in Fig. 6.3b, works as follows: 222
input: k 2 K and m = (a1 , . . . , av ) 2 X ` for some v 2 {0, . . . , `} output: a tag in K t k for i 1 to v do: t F (t, ai ) output t
The following theorem shows that F ⇤ is a prefixfree secure PRF. Theorem 6.4. Let F be a secure PRF defined over (K, X , K). Then for any polybounded value `, the cascade F ⇤ of F is a prefixfree secure PRF defined over (K, X ` , K). In particular, for every prefixfree PRF adversary A that attacks F ⇤ as in Attack Game 4.2, and issues at most Q queries, there exists a PRF adversary B that attacks F as in Attack Game 4.2, where B is an elementary wrapper around A, such that PRFpf adv[A, F ⇤ ] Q` · PRFadv[B, F ].
(6.14)
Exercise 6.6 develops an attack on fixedlength F ⇤ that demonstrates that security degrades quadratically in Q. This is disturbing as it appears to contradict the linear dependence on Q in (6.14). However, rest assured there is no contradiction here. p The adversary A from Exercise 6.6, which uses ` = 3, has advantage about 1/2 when Q is about K. Plugging A into the proof of Theorem 6.4 we obtain a PRF adversary B that attacks the PRF F making p about Q queries to gain an advantage about 1/Q. Note that 1/Q ⇡ Q/K when Q is close to K. There is nothing surprising about this adversary B: it is essentially the universal PRF attacker from Exercise 4.27. Hence, (6.14) is consistent with the attack from Exercise 6.6. Another way to view this is that the quadratic dependence on Q is already present in (6.14) because there is an implicit factor of Q hiding in the quantity PRFadv[B, F ]. The proof of Theorem 6.4 is similar to the proof that the variablelength tree construction in Section 4.6 is a prefixfree secure PRF (Theorem 4.11). Let us briefly explain how to extend the proof of Theorem 4.11 to prove Theorem 6.4. Relation to the tree construction. The cascade construction is a generalization of the variablelength tree construction of Section 4.6. Recall that the tree construction builds a secure PRF from a secure PRG that maps a seed to a pair of seeds. It is easy to see that when F is a PRF defined over (K, {0, 1}, K) then Theorem 6.4 is an immediate corollary of Theorem 4.11: simply define the PRG G mapping k 2 K to G(k) := (F (k, 0), F (k, 1)) 2 K2 , and observe that cascade applied to F is the same as the variablelength tree construction applied to G. The proof of Theorem 4.11 generalizes easily to prove Theorem 6.4 for any PRF. For example, suppose that F is defined over (K, {0, 1, 2}, K). This corresponds to a PRG G mapping k 2 K to G(k) := (F (k, 0), F (k, 1), F (k, 2)) 2 K3 . The cascade construction construction applied to F can be viewed as a ternary tree, instead of a binary tree, and the proof of Theorem 4.11 carries over with no essential changes. But why stop at width three? We can make the tree as wide as we wish. The cascade construction using a PRF F defined over (K, X , K) corresponds to a tree of width X . Again, the proof of Theorem 4.11 carries over with no essential changes. We leave the details as an exercise for the interested reader (Exercise 4.26 may be convenient here). 223
Comparing the CBC and cascade PRFs. Note that CBC uses a fixed key k for all applications of F while cascade uses a di↵erent key in each round. Since block ciphers are typically optimized to encrypt many blocks using the same key, the constant rekeying in cascade may result in worse performance than CBC. Hence, CBC is the more natural choice when using an o↵ the shelf block cipher like AES. An advantage of cascade is that there is no additive error term in Theorem 6.4. Consequently, the cascade construction remains secure even if the underlying PRF has a small domain X . CBC, in contrast, is secure only when X is large. As a result, cascade can be used to convert a PRG into a PRF for large inputs while CBC cannot.
6.4.3
Extension attacks: CBC and cascade are insecure MACs
We show that the MACs derived from CBC and cascade are insecure. This will imply that CBC and cascade are not secure PRFs. All we showed in the previous section is that CBC and cascade are prefixfree secure PRFs. Extension attack on cascade.
Given F ⇤ (k, m) for some message m in X ` , anyone can compute t0 := F ⇤ (k, m k m0 )
(6.15)
for any m0 2 X ⇤ , without knowledge of k. Once F ⇤ (k, m) is known, anyone can continue evaluating the chain using blocks of the message m0 and obtain t0 . We refer to this as the extension property of cascade. The extension property immediately implies that the MAC derived from F ⇤ is terribly insecure. The forger can request the MAC on message m and then deduce the MAC on m k m0 for any m0 of his choice. It follows, by Theorem 6.2, that F ⇤ is not a secure PRF. An attack on CBC. We describe a simple MAC forger on the MAC derived from CBC. The forger works as follows: 1. 2. 3.
pick an arbitrary a1 2 X ; request the tag t on the oneblock message (a1 ); define a2 := a1 t and output t as a MAC forgery for the twoblock message (a1 , a2 ) 2 X 2 .
Observe that t = F (k, a1 ) and a1 = F (k, a1 ) FCBC k, (a1 , a2 )
a2 . By definition of CBC we have:
= F k, F (k, a1 )
a2
= F (k, a1 = t.
Hence, (a1 , a2 ), t is an existential forgery for the MAC derived from CBC. Consequently, FCBC cannot be a secure PRF. Note that the attack on the cascade MAC is far more devastating than on the CBC MAC. But in any case, these attacks show that neither CBC nor cascade should be used directly as MACs.
6.5
From prefixfree secure PRF to fully secure PRF (method 1): encrypted PRF
We show how to convert the prefixfree secure PRFs FCBC and F ⇤ into secure PRFs, which will give us secure MACs for variable length inputs. More generally, we show how to convert a prefixfree secure PRF PF to a secure PRF. We present three methods: 224
m
k1
t2Y
PF
tag
F
k2
Figure 6.4: The encrypted PRF construction EF (k, m)
• Encrypted PRF: encrypt the short output of PF with another PRF. • Prefixfree encoding: encode the input to PF so that no input is a prefix of another. • CMAC: a more efficient prefixfree encoding using randomization. In this section we discuss the encrypted PRF method. The construction is straightforward. Let PF be a PRF mapping X ` to Y and let F be a PRF mapping Y to T . Define EF (k1 , k2 ), m := F k2 , PF (k1 , m)
(6.16)
The construction is shown in Fig. 6.4. We claim that when PF is either CBC or cascade then EF is a secure PRF. More generally, we show that EF is secure whenever PF is an extendable PRF, defined as follows: Definition 6.4. Let PF be a PRF defined over (K, X ` , Y). We say that PF is an extendable PRF if for all k 2 K, x, y 2 X ` 1 , and a 2 X we have: if
PF (k, x) = PF (k, y)
then
PF (k, x k a) = PF (k, y k a).
It is easy to see that both CBC and cascade are extendable PRFs. The next theorem shows that when PF is an extendable, prefixfree secure PRF then EF is a secure PRF. Theorem 6.5. Let PF be an extendable and prefixfree secure PRF defined over (K1 , X `+1 , Y), where Y is superpoly and ` is polybounded. Let F be a secure PRF defined over (K2 , Y, T ). Then EF , as defined in (6.16), is a secure PRF defined over (K1 ⇥ K2 , X ` , T ). In particular, for every PRF adversary A that attacks EF as in Attack Game 4.2, and issues at most Q queries, there exist a PRF adversary B1 attacking F as in Attack Game 4.2, and a prefixfree PRF adversary B2 attacking PF as in Attack Game 4.2, where B1 and B2 are elementary wrappers around A, such that PRFadv[A, EF ] PRFadv[B1 , F ] + PRFpf adv[B2 , PF ] +
Q2 . 2Y
(6.17)
We prove Theorem 6.5 in the next chapter (Section 7.3.1) after we develop the necessary tools. Note that to make EF a secure PRF on inputs of length up to `, this theorem requires that PF is prefixfree secure on inputs of length ` + 1. 225
a1
F (k1 , ·)
a2
a3
a`
L
L
L
F (k1 , ·)
F (k1 , ·)
F (k1 , ·)
···
F (k2 , ·)
tag
CBC (a) The ECBC construction ECBC(k, m)
k1
a1
a2
F
F
(encrypted CBC)
a`
···
F
cascade
t2K
t k fpad
k2
(b) The NMAC construction NMAC(k, m)
F
tag
(encrypted cascade)
Figure 6.5: Secure PRF constructions for variable length inputs The bound in (6.17) is tight. Although not entirely necessary, let us assume that Y = T , that F is a block cipher, and that X  is not too small. These assumptions will greatlypsimplify the argument. We exhibit an attack that breaks EF with constant probability after Q ⇡ Y queries. Our attack will, in fact, break EF as a MAC. The adversary picks Q random inputs x1 , . . . , xQ 2 X 2 and queries its MAC challenger at all Q inputs to obtain t1 , . . . , tQ 2 T . By the birthday paradox (Corollary B.2), for any fixed key k1 , with constant probability there will be distinct indices i, j such that xi 6= xj and PF (k1 , xi ) = PF (k1 , xj ). On the one hand, if such a collision occurs, we will detect it, because ti = tj for such a pair of indices. On the other hand, if ti = tj for some pair of indices i, j, then our assumption that F is a block cipher guarantees that PF (k1 , xi ) = PF (k1 , xj ). Now, assuming that xi 6= xj and PF (k1 , xi ) = PF (k1 , xj ), and since PF is extendable, we know that for all a 2 X , we have PF k1 , (xi k a) = PF k1 , (xj k a) . Therefore, our adversary can obtain the MAC tag t for xi k a, and this tag t will also be a valid tag for xj k a. This attack easily generalizes to show the necessity of the term Q2 /(2Y) in (6.17).
6.5.1
ECBC and NMAC: MACs for variable length inputs
Figures 6.5a and 6.5b show the result of applying the EF construction (6.16) to CBC and cascade.
226
The EncryptedCBC PRF Applying EF to CBC results in a classic PRF (and hence a MAC) called encryptedCBC or ECBC for short. This MAC is standardized by ANSI (see Section 6.9) and is used in the banking industry. The ECBC PRF uses the same underlying PRF F for both CBC and the final encryption. Consequently, ECBC is defined over (K2 , X ` , X ). Theorem 6.6 (ECBC security). Let F be a secure PRF defined over (K, X , X ). Suppose X is superpoly, and let ` be a polybounded length parameter. Then ECBC is a secure PRF defined over (K2 , X ` , X ). In particular, for every PRF adversary A that attacks ECBC as in Attack Game 4.2, and issues at most Q queries, there exist PRF adversaries B1 , B2 that attack F as in Attack Game 4.2, and which are elementary wrappers around A, such that PRFadv[A, ECBC] PRFadv[B1 , F ] + PRFadv[B2 , F ] +
(Q(` + 1))2 + Q2 . 2X 
(6.18)
Proof. CBC is clearly extendable and is a prefixfree secure PRF by Theorem 6.3. Hence, if the underlying PRF F is secure, then ECBC is a secure PRF by Theorem 6.5. 2 p The argument given after Theorem 6.5 shows that there is an attacker that after Q ⇡ X  queries breaks this PRF with constant advantage. Recall that for 3DES we have X = {0, 1}64 . Hence, after about a billion queries (or more precisely, 232 queries) an attacker can break the ECBC3DES MAC with constant probability. The NMAC PRF Applying EF to cascade results in a PRF (and hence a MAC) called Nested MAC or NMAC for short. A variant of this MAC is standardized by the IETF (see Section 8.7.2) and is widely used in Internet protocols. We wish to use the same underlying PRF F for the cascade construction and for the final encryption. Unfortunately, the output of cascade is in K while the message input to F is in X . To solve this problem we need to embed the output of cascade into X . More precisely, we assume that K X  and that there is an efficiently computable onetoone function g that maps K into X . For example, suppose K := {0, 1} and X := {0, 1}n where n. Define g(t) := t k fpad where fpad is a fixed pad of length n bits. This fpad can be as simple as a string of 0s. With this translation, all of NMAC can be built from a single secure PRF F , as shown in Fig. 6.5b. Theorem 6.7 (NMAC security). Let F be a secure PRF defined over (K, X , K), where K can be embedded into X . Then NMAC is a secure PRF defined over (K2 , X ` , K). In particular, for every PRF adversary A that attacks NMAC as in Attack Game 4.2, and issues at most Q queries, there exist PRF adversaries B1 , B2 that attack F as in Attack Game 4.2, and which are elementary wrappers around A, such that PRFadv[A, NMAC] (Q(` + 1)) · PRFadv[B1 , F ] + PRFadv[B2 , F ] +
Q2 . 2K
(6.19)
Proof. NMAC is clearly extendable and is a prefixfree secure PRF by Theorem 6.4. Hence, if the underlying PRF F is secure, then NMAC is a secure PRF by Theorem 6.5. 2 227
ECBC and NMAC are streaming MACs. Both ECBC and NMAC can be used to authenticate variable size messages in X ` . Moreover, there is no need for the message length to be known ahead of time. A MAC that has this property is said to be a streaming MAC. This property enables applications to feed message blocks to the MAC one block at a time and at some arbitrary point decide that the message is complete. This is important for applications like streaming video, where the message length may not be known ahead of time. In contrast, some MAC systems require that the message length be prepended to the message body (see Section 6.6). Such MACs are harder to use in practice since they require applications to determine the message length before starting the MAC calculations.
6.6
From prefixfree secure PRF to fully secure PRF (method 2): prefixfree encodings
Another approach to converting a prefixfree secure PRF into a secure PRF is to encode the input to the PRF so that no encoded input is a prefix of another. We use the following terminology: • We say that a set S ✓ X ` is a prefixfree set if no element in S is a proper prefix of any other. For example, if (x1 , x2 , x3 ) belongs to a prefixfree set S, then neither x1 nor (x1 , x2 ) are in S. • Let X ` denote the set of all nonempty strings over X of length at most `. We say that a function pf : M ! X ` is a prefixfree encoding if pf is injective (i.e., onetoone) and the image of pf in is a prefixfree set. >0
>0
Let PF be a prefixfree secure PRF defined over (K, X ` , Y) and pf : M ! X ` be a prefixfree encoding. Define the derived PRF F as >0
F (k, m) := PF (k, pf (m)). Then F is defined over (K, M, Y). We obtain the following trivial theorem. Theorem 6.8. If PF is a prefixfree secure PRF and pf is a prefixfree encoding then F is a secure PRF.
6.6.1
Prefix free encodings
To construct PRFs using Theorem 6.8 we describe two prefixfree encodings pf : M ! X ` . We assume that X = {0, 1}n for some n. Method 1: prepend length.
Set M := X `
1
and let m = (a1 , . . . , av ) 2 M. Define
pf (m) := (hvi, a1 , . . . , av )
2 X ` >0
where hvi 2 X is the binary representation of v, the length of m. We assume that ` < 2n so that the message length can be encoded as an nbit binary string. We argue that pf is a prefixfree encoding. Clearly pf is injective. To see that the image of pf is a prefixfree set let pf (x) and pf (y) be two elements in the image of pf . If pf (x) and pf (y) contain the same number of blocks, then neither is a proper prefix of the other. Otherwise, pf (x) 228
and pf (y) contain a di↵erent number of blocks and must therefore di↵er in the first block. But then, again, neither is a proper prefix of the other. Hence, pf is a prefixfree encoding. This prefixfree encoding is not often used in practice since the resulting MAC is not a streaming MAC: an application using this MAC must commit to the length of the message to MAC ahead of time. This is undesirable for streaming applications such as streaming video where the length of packets may not be known ahead of time. Method 2: stop bits.
Let X¯ := {0, 1}n
1
and let M = X¯ ` . For m = (a1 , . . . , av ) 2 M, define >0
pf (m) := (a1 k 0), (a2 k 0), . . . , (av
1
k 0), (av k 1)
2 X ` >0
Clearly pf is injective. To see that the image of pf is a prefixfree set let pf (x) and pf (y) be two elements in the image of pf . Let v be the number of blocks in pf (x). If pf (y) contains v or fewer blocks then pf (x) is not a proper prefix of pf (y). If pf (y) contains more than v blocks then block number v in pf (y) ends in 0, but block number v in pf (x) ends in 1. Hence, pf (x) and pf (y) di↵er in block v and therefore pf (x) is not a proper prefix of pf (y). The MAC resulting from this prefixfree encoding is a streaming MAC. This encoding, however, increases the length of the message to MAC by v bits. When computing the MAC on a long message using either CBC or cascade, this encoding will result in additional evaluations of the underlying PRF (e.g. AES). In contrast, the encrypted PRF method of Section 6.5 only adds one additional application of the underlying PRF. For example, to MAC a megabyte message (220 bytes) using ECBCAES and pf one would need an additional 511 evaluations of AES beyond what is needed for the encrypted PRF method. In practice, things are even worse. Since computers prefer bytealigned data, one would most likely need to append an entire byte to every block, rather than just a bit. Then to MAC a megabyte message using ECBCAES and pf would result in 4096 additional evaluations of AES over the encrypted PRF method — an overhead of about 6%.
6.7
From prefixfree secure PRF to fully secure PRF (method 3): CMAC
Both prefix free encoding methods from the previous section are problematic. The first resulted in a nonstreaming MAC. The second required more evaluations of the underlying PRF for long messages. We can do better by randomizing the prefix free encoding. We build a streaming secure PRF that introduces no overhead beyond the underlying prefixfree secure PRF. The resulting MACs, shown in Fig. 6.6, are superior to those obtained from encrypted PRFs and deterministic encodings. This approach is used in a NIST MAC standard called CMAC and described in Section 6.10. First, we introduce some convenient notation: Definition 6.5. For two strings x, y 2 X ` , let us write x ⇠ y if x is a prefix of y or y is a prefix of x. Definition 6.6. Let ✏ be a real number, with 0 ✏ 1. A randomized ✏prefixfree encoding is a function rpf : K ⇥ M ! X ` such that for all m0 , m1 2 M with m0 6= m1 , we have ⇥ ⇤ Pr rpf (k, m0 ) ⇠ rpf (k, m1 ) ✏, >0
where the probability is over the random choice of k in K. 229
Note that the image of rpf (k, ·) need not be a prefixfree set. However, without knowledge of k it is difficult to find messages m0 , m1 2 M such that rpf (k, m0 ) is a proper prefix of rpf (k, m1 ) (or vice versa). The function rpf (k, ·) need not even be injective. A simple rpf .
Let K := X and M := X ` . Define >0
rpf (k, (a1 , . . . , av )) := a1 , . . . , av
1 , (av
k) 2 X ` >0
It is easy to see that rpf is a randomized (1/X )prefixfree encoding. Let m0 , m1 2 M with m0 6= m1 . Suppose that m0  = m1 . Then it is clear that for all choices of k, rpf (k, m0 ) and rpf (k, m1 ) are distinct strings of the same length, and so neither is a prefix of the other. Next, suppose that m0  < m1 . If v := rpf (k, m0 ), then clearly rpf (k, m0 ) is a proper prefix of rpf (k, m1 ) if and only if m0 [v 1] k = m1 [v 1]. But this holds with probability 1/X  over the random choice of k, as required. Finally, the case m0  > m1  is handled by a symmetric argument. Using rpf . Let PF be a prefixfree secure PRF defined over (K, X ` , Y) and rpf : K1 ⇥M ! X ` be a randomized prefixfree encoding. Define the derived PRF F as >0
F (k, k1 ), m) := PF k, rpf (k1 , m) .
(6.20)
Then F is defined over (K ⇥ K1 , M, Y). We obtain the following theorem, which is analogous to Theorem 6.8. Theorem 6.9. If PF is a prefixfree secure PRF, ✏ is negligible, and rpf a randomized ✏prefixfree encoding, then F defined in (6.20) is a secure PRF. In particular, for every PRF adversary A that attacks F as in Attack Game 4.2, and issues at most Q queries, there exist prefixfree PRF adversaries B1 and B2 that attack PF as in Attack Game 4.2, where B1 and B2 are elementary wrappers around A, such that PRFadv[A, F ] PRFpf adv[B1 , PF ] + PRFpf adv[B2 , PF ] + Q2 ✏/2.
(6.21)
Proof idea. If the adversary’s set of inputs to F give rise to a prefixfree set of inputs to PF , then the adversary sees just some random looking outputs. Moreover, if the adversary sees random outputs, it obtains no information about the rpf key k1 , which ensures that the set of inputs to PF is indeed prefix free (with overwhelming probability). Unfortunately, this argument is circular. However, we will see in the detailed proof how to break this circularity. 2 Proof. Without loss of generality, we assume that A never issues the same query twice. We structure the proof as a sequence of three games. For j = 0, 1, 2, we let Wj be the event that A outputs 1 at the end of Game j. Game 0. The challenger in Experiment 0 of the PRF Attack Game 4.2 with respect to F works as follows.
230
k
R
K,
k1
R
K1
upon receiving a signing query mi 2 M (for i = 1, 2, . . .) do: rpf (k1 , mi ) 2 X ` xi yi PF (k, xi ) send yi to A >0
Game 1. We change the challenger in Game 0 to ensure that all queries to PF are prefix free. Recall the notation x ⇠ y, which means that x is a prefix of y or y is a prefix of x. k
R
K,
k1
R
K1 ,
r1 , . . . , rQ
R
Y
upon receiving a signing query mi 2 M (for i = 1, 2, . . .) do: rpf (k1 , mi ) 2 X ` xi (1) if xi ⇠ xj for some j < i ri then yi (2) else yi PF (k, xi ) send yi to A >0
Let Z1 be the event that the condition on line (1) holds at some point during Game 1. Clearly, Games 1 and 2 proceed identically until event Z1 occurs; in particular, W0 ^ Z¯1 occurs if and only if W1 ^ Z¯1 occurs. Applying the Di↵erence Lemma (Theorem 4.7), we obtain Pr[W1 ]
Pr[W0 ] Pr[Z1 ].
(6.22)
Unfortunately, we are not quite in a position to bound Pr[Z1 ] at this point. At this stage in the analysis, we cannot say that the evaluations of PF at line (2) do not leak some information about k1 that could help A make Z1 happen. This is the circularity problem we alluded to above. To overcome this problem, we will delay the analysis of Z1 to the next game. Game 2. Now we play the usual “PRF card,” replacing the function PF (k, ·) by a truly random function. This is justified, since by construction, in Game 1, the set of inputs to PF (k, ·) is prefixfree. To implement this change, we may simply replace the line marked (2) by (2)
else yi
ri
After making this change, we see that yi gets assigned the random value ri , regardless of whether the condition on line (1) holds or not. Now, let Z2 be the event that the condition on line (1) holds at some point during Game 2. It is not hard to see that (6.23) Pr[Z1 ] Pr[Z2 ] PRFpf adv[B1 , F ] and Pr[W1 ]
Pr[W2 ] PRFpf adv[B2 , F ]
(6.24)
for efficient prefixfree PRF adversaries B1 and B2 . These two adversaries are basically the same, except that B1 outputs 1 if the condition on line (1) holds, while B2 ouputs whatever A outputs. Moreover, in Game 2, the value of k1 is clearly independent of A’s queries, and so by making use of the ✏prefixfree property of rpf , and the union bound we have Pr[Z2 ] Q2 ✏/2 231
(6.25)
a1
F (k, ·)
a2
a3
a`
L
L
L
F (k, ·)
F (k, ·)
F (k, ·)
···
k1
tag
(a) rpf applied to CBC
a1
a2
a3
···
a` L
k
F
F
F
F
k1
tag
(b) rpf applied to cascade
Figure 6.6: Secure PRFs using random prefixfree encodings Finally, Game 2 perfectly emulates for A a random function in Funs[M, Y]. Game 2 is therefore identical to Experiment 1 of the PRF Attack Game 4.2 with respect to F , and hence Pr[W0 ]
Pr[W2 ] = PRFadv[A, F ].
(6.26)
Now combining (6.22)–(6.26) proves the theorem. 2
6.8
Converting a blockwise PRF to bitwise PRF
So far we constructed a number of PRFs for variable length inputs in X ` . Typically X = {0, 1}n where n is the block size of the underlying PRF from which CBC or cascade are built (e.g., n = 128 for AES). All our MACs so far are designed to authenticate messages whose length is a multiple of n bits. In this section we show how to convert these PRFs into PRFs for messages of arbitrary bit length. That is, given a PRF for messages in X ` we construct a PRF for messages in {0, 1}n` . Let F be a PRF taking inputs in X `+1 . Let inj : {0, 1}n` ! X `+1 be an injective (i.e., onetoone) function. Define the derived PRF Fbit as Fbit (k, x) := F (k, inj (x)). Then we obtain the following trivial theorem. Theorem 6.10. If F is a secure PRF defined over (K, X `+1 , Y) then Fbit is a secure PRF defined over (K, {0, 1}n` , Y). 232
case 1:
a1
case 2:
a1
a2
a2
!
a1
!
a1
a2
1000
a2
1000000
Figure 6.7: An injective function inj : {0, 1}n` ! X `+1 An injective function. For X := {0, 1}n , a standard example of an injective inj from {0, 1}n` to X `+1 works as follows. If the input message length is not a multiple of n then inj appends 100 . . . 00 to pad the message so its length is the next multiple of n. If the given message length is a multiple of n then inj appends an entire nbit block (1 k 0n 1 ). Fig. 6.7 describes this in a picture. More precisely, the function works as follows: input: m 2 {0, 1}n`
u m mod n, m0 m k 1 k 0n u 1 output m0 as a sequence of nbit message blocks To see that inj is injective we show that it is invertible. Given y inj (m) scan y from right to left and remove all the 0s until and including the first 1. The remaining string is m. A common mistake is to pad the given message to a multiple of a block size using an all0 pad. This pad is not injective and results in an insecure MAC: for any message m whose length is not a multiple of the block length, the MAC on m is also a valid MAC for m k 0. Consequently, the MAC is vulnerable to existential forgery. Injective functions must expand. When we feed an nbit single block message into inj , the function adds a “dummy” block and outputs a twoblock message. This is unfortunate for applications that MAC many single block messages. When using CBC or cascade, the dummy block forces the signer and verifier to evaluate the underlying PRF twice for each message, even though all messages are one block long. Consequently, inj forces all parties to work twice as hard as necessary. It is natural to look for injective functions from {0, 1}n` to X ` that never add dummy blocks. Unfortunately, there are no such functions simply because the set {0, 1}n` is larger than the set X ` . Hence, all injective functions must occasionally add a “dummy” block to the output. The CMAC construction described in Section 6.10 provides an elegant solution to this problem. CMAC avoids adding dummy blocks by using a randomized injective function.
6.9
Case study: ANSI CBCMAC
When building a MAC from a PRF, implementors often shorten the final tag by only outputting the w most significant bits of the PRF output. Exercise 4.4 shows that truncating a secure PRF has no e↵ect on its security as a PRF. Truncation, however, a↵ects the derived MAC. Theorem 6.2 shows that the smaller w is the less secure the MAC becomes. In particular, the theorem adds a 1/2w error in the concrete security bounds. Two ANSI standards (ANSI X9.9 and ANSI X9.19) and two ISO standards (ISO 87311 and ISO/IEC 9797) specify variants of ECBC for message authentication using DES as the underlying 233
PRF. These standards truncate the final 64bit output of the ECBCDES and use only the leftmost w bits of the output, where w = 32, 48, or 64 bits. This reduces the tag length at the cost of reduced security. Both ANSI CBCMAC standards specify a padding scheme to be used for messages whose length is not a multiple of the DES or AES block size. The padding scheme is identical to the function inj described in Section 6.8. The same padding scheme is used when signing a message and when verifying a messagetag pair.
6.10
Case study: CMAC
Cipherbased MAC — CMAC — is a variant of ECBC adopted by the National Institute of Standards (NIST) in 2005. It is based on a proposal due to Black and Rogaway and an extension due to Iwata and Kurosawa. CMAC improves over ECBC used in the ANSI standard in two ways. First, CMAC uses a randomized prefixfree encoding to convert a prefixfree secure PRF to a secure PRF. This saves the final encryption used in ECBC. Second, CMAC uses a “two key” method to avoid appending a dummy message block when the input message length is a multiple of the underlying PRF block size. CMAC is the best approach to building a bitwise secure PRF from the CBC prefixfree secure PRF. It should be used in place of the ANSI method. In Exercise 6.14 we show that the CMAC construction applies equally well to cascade. The CMAC bitwise PRF. The CMAC algorithm consists of two steps. First, a subkey generation algorithm is used to derive three keys k0 , k1 , k2 from the MAC key k. Then the three keys k0 , k1 , k2 are used to compute the MAC. Let F be a PRF defined over (K, X , X ) where X = {0, 1}n . The NIST standard uses AES as the PRF F . The CMAC signing algorithm is given in Table 6.1 and is illustrated in Fig. 6.8. The figure on the left is used when the message length is a multiple of the block size n. The figure on the right is used otherwise. The standard allows for truncating the final output to w bits by only outputting the w most significant bits of the final value t. Security. The CMAC algorithm described in Fig. 6.8 can be analyzed using the randomized prefixfree encoding paradigm. In e↵ect, CMAC converts the CBC prefixfree secure PRF directly into a bitwise secure PRF using a randomized prefixfree encoding rpf : K ⇥ M ! X ` where K := X 2 and M := {0, 1}n` . The encoding rpf is defined as follows: >0
input: m 2 M and (k1 , k2 ) 2 X 2
if m is not a positive multiple of n then u m mod n partition m into a sequence of bit strings a1 , . . . , av 2 X , so that m = a1 k · · · k av and a1 , . . . , av 1 are nbit strings
if m is a positive multiple of n then output a1 , . . . , av 1 , (av k1 ) else output a1 , . . . , av 1 , ((av k 1 k 0n
u 1)
k2 )
The argument that rpf is a randomized 2 n prefixfree encoding is similar to the one is Section 6.7. Hence, CMAC fits the randomized prefixfree encoding paradigm and its security follows from 234
input: Key k 2 K and m 2 {0, 1}⇤ output: tag t 2 {0, 1}w for some w n
Setup: Run a subkey generation algorithm to generate keys k0 , k1 , k2 2 X from k 2 K ` length(m) u max(1, d`/ne) Break m into consecutive nbit blocks so that m = a1 k a2 k · · · k au 1 k a⇤u where a1 , . . . , au ⇤ (⇤) If length(au ) = n then au = k1 a⇤u else au = k2 (a⇤u k 1 k 0j ) where j = nu ` 1 CBC: t 0n for i 1 to u do: t F (k0 , t Output t[0 . . . w
1
2 {0, 1}n .
ai ) 1]
//
Output w most significant bits of t.
Table 6.1: CMAC signing algorithm
(a) when length(m) is a positive multiple of n a1
F (k, ·)
a2
···
(b) otherwise
au
L
L
F (k, ·)
F (k, ·)
a1 k1
F (k, ·)
tag
a2
···
au k100
L
L
F (k, ·)
F (k, ·)
tag
Figure 6.8: CMAC signing algorithm
235
k2
Theorem 6.9. The keys k1 , k2 are used to resolve collisions between a message whose length is a positive multiple of n and a message that has been padded to make it a positive multiple of n. This is essential for the analysis of the CMAC rpf . Subkey generation. The subkey generation algorithm generates the keys (k0 , k1 , k2 ) from k. It uses a fixed mask string Rn that depends on the block size of F . For example, for a 128bit block size, the standard specifies R128 := 0120 10000111. For a bit string X we denote by X 0
Note that H is certainly not a secure PRF, even if we restrict ourselves to nonadaptive or prefixfree adversaries: given H(k, m) for any message m, we can efficiently compute the key k. 7.18 (Optimal collision probability with shorter hash keys). For positive integer d, let Id := {0, . . . , d 1} and Id⇤ := {1, . . . , d 1}. (a) Let N be a positive integer and p be a prime. Consider the keyed hash function H defined over (Ip ⇥ Ip⇤ , Ip , IN ) as follows: H((k0 , k1 ), a) := ((k0 + ak1 ) mod p) mod N . Show that H is a 1/N UHF. (b) While the construction in part (a) gives a UHF with “optimal” collision probability, the key space is unfortunately larger than the message space. Using the result of part (a), along with part (a) of Exercise 7.15 and the result of Exercise 7.16, you are to design a hash function with “nearly optimal” collision probability, but with much smaller keys. Let N and ` be positive integers. Let ↵ be a number with 0 < ↵ < 1. Design a (1 + ↵)/N UHF with message space {0, 1}` and output space IN , where keys bit strings of length O(log(N `/↵)). 7.19 (Inner product hash). Let p be a prime. (a) Consider the keyed hash function H defined over (Z`p , Z`p , Zp ) as follows: H((k1 , . . . , k` ), (a1 , . . . , a` )) := a1 k1 + · · · + a` k` . Show that H is a 1/pDUF. (b) Since multiplications can be much more expensive than additions, the following variant of the hash function in part (a) is sometimes preferable. Assume ` is even, and consider the keyed
274
hash function H 0 defined over (Z`p , Z`p , Zp ) as follows: 0
H ((k1 , . . . , k` ), (a1 , . . . , a` )) :=
`/2 X
(a2i
1
+ k2i
1 )(a2i
+ k2i ).
i=1
Show that H 0 is also a 1/pDUF. (c) Although both H and H 0 are ✏DUFs with “optimal” ✏ values, the keys are unfortunately very large. Using a similar approach to part (b) of the previous exercise, design a (1 + ↵)/pDUF with message space {0, 1}` and output space Zp , where keys bit strings of length O(log(p`/↵)). 7.20 (Divisionfree hash). This exercise develops a hash function that does not require and division or mod operations, which can be expensive. It can be implemented just using shifts and adds. For positive integer d, let Id := {0, . . . , d 1}. Let n be a positive integer and set N := 2n . ` , I ` , Z ) as follows: (a) Consider the keyed hash function H defined over (IN 2 N N
H((k1 , . . . , k` ), (a1 , . . . , a` )) := [t]N 2 ZN , where t :=
⌅
X
ai ki mod N 2
i
⇧ N .
Show that H is a 2/N DUF. Below in Exercise 7.30 we will see a minor variant of H that satisfies a stronger property, and in particular, is a 1/N DUF. (b) Analogous to part (b) in the previous exercise, assume ` is even, and consider the keyed hash ` , I ` , Z ) as follows: function H defined over (IN 2 N N H 0 ((k1 , . . . , k` ), (a1 , . . . , a` )) := [t]N 2 ZN , where t := Show that
H0
⌅
`/2 X
(a2i
1
+ k2i
1 )(a2i
i=1
is a 2/N DUF.
+ k2i ) mod N 2
⇧ N .
7.21 (DUF to UHF conversion). Let H be a keyed hash function defined over (K, M, ZN ). We construct a new keyed hash function H 0 , defined over (K, M ⇥ ZN , ZN ) as follows: H 0 (k, (m, x)) := H(k, m) + x. Show that if H is an ✏DUF, then H 0 is an ✏UHF. 7.22 (DUF modulus switching). We will be working with DUFs with digest spaces Zm for various m, and so to make things clearer, we will work with digest spaces that are plain old sets of integers, and state explicitly the modulus m, as in “an ✏DUF modulo m”. For positive integer d, let Id := {0, . . . , d 1}. Let p and N be integers greater than 1. Let H be a keyed hash function defined over (K, M, Ip ). Let H 0 be the keyed hash function defined over (K, M, IN ) as follows: H 0 (k, m) := H(k, m) mod N . (a) Show that if p N/2 and H is an ✏DUF modulo p, then H 0 is an ✏DUF modulo N .
(b) Suppose that p N and H is an ✏DUF modulo p. Show that H 0 is an ✏0 DUF modulo N for ✏0 = 2(p/N + 1)✏. In particular, if ✏ = ↵/p, we can take ✏0 = 4↵/N . 275
7.23 (More flexible output spaces). As in the previous exercise, we work with DUFs whose digest spaces are plain old sets of integers, but we explicitly state the modulus m. Again, for positive integer d, we let Id := {0, . . . , d 1}. Let 1 < N p, where p is prime.
⇤ ` , I ) as follows: is the keyed hash function defined over (Ip , IN (a) Hfxpoly N ✓ ◆ ⇤ Hfxpoly (k, (a1 , . . . , a` )) := (a1 k ` + · · · + a` k mod p mod N. ⇤ Show that Hfxpoly is a 4`/N DUF modulo N .
` ⇤ is the keyed hash function defined over (Ip , IN , IN ) as follows: (b) Hxpoly ✓ ◆ ⇤ v+1 v + a1 k + · · · + av k mod p mod N. Hxpoly (k, (a1 , . . . , av )) := (k ⇤ is a 4(` + 1)/N DUF modulo N . Show that Hxpoly ⇤ ` , I ) as follows: is the keyed hash function defined over (Ip , IN (c) Hfpoly N ◆ ✓ ◆ ⇤ ` 1 Hfpoly (k, (a1 , . . . , a` )) := (a1 k + · · · + a` 1 k mod p + a` mod N. ⇤ Show that Hfpoly is a 4(`
1)/N UHF.
` ⇤ is the keyed hash function is defined over (Ip , IN , IN ) as follows: (d) Hpoly ◆ ✓ ◆ ⇤ v v 1 + · · · + av 1 k mod p + av mod N. Hpoly (k, (a1 , . . . , av )) := (k + a1 k ⇤ for v > 0, and for zerolength messages, it is defined to be the constant 1. Show that Hpoly is a 4`/N UHF.
Hint: All of these results follow easily from the previous two exercises, except that the analysis in part (d) requires that zerolength messages are treated separately. 7.24 (Be careful: reducing at the wrong time can be dangerous). With notation as in the previous exercise, show that if (3/2)N p < 2N , the keyed hash function H defined over 2 , I ) as (Ip , IN N H(k, (a, b)) := ((ak + b) mod p) mod N is not a (1/3)UHF. Contrast this function with that in part (c) of the previous exercise with ` = 2. 7.25 (A PMAC0 alternative). Again, for positive integer d, let Id := {0, . . . , d 1}. Let N = 2n and let p be a prime with N/4 < p < N/2. Let H be the hash function defined over (IN/4 , IN ⇥ IN/4 , IN ) as follows: H(k, (a, i)) := (((i · k) mod p) + a) mod N. (a) Show that H is a 4/N UHF. Hint: Use Exercise 7.21 and part (a) of Exercise 7.22. 276
(b) Show how to use H to modify PMAC0 so that the message space is Y ` (where Y = {0, 1}n and ` < N/4), and the PRF F1 is defined over (K1 , Y, Y). Analyze the security of your construction, giving a concrete security bound. 7.26 (Collision lowerbounds for Hpoly ). Consider the function Hpoly (k, m) defined in (7.3) using a prime p and assume ` = 2. (a) Show that for all sufficiently large p, the following holds: for any fixed k 2 Zp , among p b pc random inputs to Hpoly (k, ·), the probability of a collision is bounded from below by a constant. Hint: Use the birthday paradox (Appendix B.1). (b) Show that given any collision for Hpoly under key k, we can efficiently compute k. That is, give an efficient algorithm that takes two inputs m, m0 2 Z2p , and that outputs kˆ 2 Zp , and satisfies the following property: for every k 2 Zp , if H(k, m) = H(k, m0 ), then kˆ = k. 7.27 (XORhash analysis). Generalize Theorem 7.6 to show that for every Qquery UHF adversary A, there exists a PRF adversary B, which is an elementary wrapper around A, such that MUHFadv[A, F ] PRFadv[B, F ] +
Q2 . 2Y
Moreover, B makes at most Q` queries to F . 7.28 (Hxpoly is not a good PUF). Show that Hxpoly defined in (7.23) is not a good PUF by exhibiting an adversary that wins Attack Game 7.5 with probability 1. 7.29 (Converting a onetime MAC to a MAC). Suppose I = (S, V ) is a (possibly randomized) MAC defined over (K1 , M, T ), where T = {0, 1}n , that is onetime secure (see Section 7.6). Further suppose that F is a secure PRF defined over (K2 , R, T ), where R is superpoly. Consider the MAC I 0 = (S 0 , V 0 ) defined over (K1 ⇥ K2 , M, R ⇥ T ) as follows: S 0 ((k1 , k2 ), m) := V 0 ((k1 , k2 ), m, (r, t0 )) :=
r t
R
R; t
R
F (k2 , r)
S(k1 , m); t0
F (k2 , r)
t; output (r, t0 )
t0 ; output V (k1 , m, t)
Show that I 0 is a secure (many time) MAC. 7.30 (Pairwise independent functions). In this exercise, we develop the notion of a PRF that is unconditionally secure, provided the adversary can make at most two queries. We say that a PRF F defined over (K, X , Y) is an ✏almost pairwise independent function, or ✏APIF, if the following holds: for all adversaries A (even inefficient ones) that make at most 2 queries in Attack Game 4.2, we have PRFadv[A, F ] ✏. If ✏ = 0, we call F a pairwise independent function, or PIF. (a) Suppose that X  > 1 and that for all x0 , x1 2 X with x0 6= x1 , and all y0 , y1 2 Y, we have Pr[F (k, x0 ) = y0 ^ F (k, x1 ) = y1 ] =
1 , Y 2
where the probability is over the random choice of k 2 K. Show that F is a PIF. 277
(b) Consider the function H 0 built from H in (7.32). Show that if H is a 1/N DUF, then H 0 is a PIF. (c) For positive integer d, let Id := {0, . . . , d 1}. Let n be a positive integer and set N := 2n . `+1 ` Consider the keyed hash function H defined over (IN 2 , IN , IN ) as follows: H((k0 , k1 , . . . , k` ), (a1 , . . . , a` )) :=
⌅
k0 +
X i
ai ki mod N 2
⇧ N .
Show that H is a PIF. Note: on a typical computer, if n is not too large, this can be implemented very easily with just integer multiplications, additions, and shifts. (d) Show that in the PRF(UHF) composition, if H is an ✏1 UHF and F is an ✏2 APIF, then the composition F 0 is an (✏1 + ✏2 )APIF. (e) Show that any ✏APIF is an (✏ + 1/Y)PUF. (f) Using an appropriate APIF, show how to construct a probabilistic cipher that is unconditionally CPA secure provided the adversary can make at most two queries in Attack Game 5.2.
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Chapter 8
Message integrity from collision resistant hashing In the previous chapter we discussed universal hash functions (UHFs) and showed how they can be used to construct MACs. Recall that UHFs are keyed hash functions for which finding collisions is difficult, as long as the key is kept secret. In this chapter we study keyless hash functions for which finding collisions is difficult. Informally, a keyless function is an efficiently computable function whose description is fully public. There are no secret keys and anyone can evaluate the function. Let H be a keyless hash function from some large message space M into a small digest space T . As in the previous chapter, we say that two messages m0 , m1 2 M are a collision for the function H if H(m0 ) = H(m1 )
and
m0 6= m1 .
Informally, we say that the function H is collision resistant if finding a collision for H is difficult. Since the digest space T is much smaller than M, we know that many such collisions exist. Nevertheless, if H is collision resistant, actually finding a pair m0 , m1 that collide should be difficult. We give a precise definition in the next section. In this chapter we will construct collision resistant functions and present several applications. To give an example of a collision resistant function we mention a US federal standard called the Secure Hash Algorithm Standard or SHA for short. The SHA standard describes a number of hash functions that o↵er varying degrees of collision resistance. For example, SHA256 is a function that hashes long messages into 256bit digests. It is believed that finding collisions for SHA256 is difficult. Collision resistant hash functions have many applications. We briefly mention two such applications here and give the details later on in the chapter. Many other applications are described throughout the book. Extending cryptographic primitives. An important application for collision resistance is its ability to extend primitives built for short inputs to primitives for much longer inputs. We give a MAC construction as an example. Suppose we are given a MAC system I = (S, V ) that only authenticates short messages, say messages that are 256 bits long. We want to extend the domain of the MAC so that it can authenticate much longer inputs. Collision resistant hashing gives a very simple solution. To compute a MAC for some long message m we first hash m and then apply S to 279
k
m
H
S
t
Figure 8.1: HashthenMAC construction the resulting short digest, as described in Fig. 8.1. In other words, we define a new MAC system I = (S 0 , V 0 ) where S 0 (k, m) := S(k, H(m)). MAC verification works analogously by first hashing the message and then verifying the tag of the digest. Clearly this hashthenMAC construction would be insecure if it were easy to find collisions for H. If an adversary could find two long messages m0 and m1 such that H(m0 ) = H(m1 ) then he could forge tags using a chosen message attack. Suppose m0 is an innocuous message while m1 is evil, say a virus infected program. The adversary would ask for the tag on the message m0 and obtain a tag t in response. Then the pair (m0 , t) is a valid messagetag pair, but so is the pair (m1 , t). Hence, the adversary is able to forge a tag for m1 , which breaks the MAC. Even worse, the valid tag may fool a user into running the virus. This argument shows that collision resistance is necessary for this hashthenMAC construction to be secure. Later on in the chapter we prove that collision resistance is, in fact, sufficient to prove security. The hashthenMAC construction looks similar to the PRF(UHF) composition discussed in the previous chapter (Section 7.3). These two methods build similar looking MACs from very di↵erent building blocks. The main di↵erence is that a collision resistant hash can extend the input domain of any MAC. On the other hand, a UHF can only extend the domain of a very specific type of MAC, namely a PRF. This is illustrated further in Exercise 7.4. Another di↵erence is that the secret key in the hashthenMAC method is exactly the same as in the underlying MAC. The PRF(UHF) method, in contrast, extends the secret key of the underlying PRF by adding a UHF secret key. The hashthenMAC construction performs better than PRF(UHF) when we wish to compute the tag for a single message m under multiple keys k1 , . . . , kn . That is, we wish to compute S 0 (ki , m) for all i = 1, . . . , n. This comes up, for example, when providing integrity for a file on disk that is readable by multiple users. The file header contains one integrity tag per user so that each user can verify integrity using its own MAC key. With the hashthenMAC construction it suffices to compute H(m) once and then quickly derive the n tags from this single hash. With a PRF(UHF) MAC, the UHF depends on the key ki and consequently we will need to rehash the entire message n times, once for each user. See also Exercise 6.4 for more on this problem. File integrity. Another application for collision resistance is file integrity also discussed in the introduction of Chapter 6. Consider a set of n critical files that change infrequently, such as certain operating system files. We want a method to verify that these files are not modified by some malicious code or malware. To do so we need a small amount of readonly memory, namely memory that the malware can read, but cannot modify. Readonly memory can be implemented, for example, using a small USB disk that has a physical switch flipped to the “readonly” position. We place a hash of each of the n critical files in the readonly memory so that this storage area only 280
Readonly memory
Disk File F1
hash file FH H(F1 )
File F2
H(F2 )
H(FH )
H(F3 ) File F3
Figure 8.2: File integrity using small readonly memory contains n short hashes. We can then check integrity of a file F by rehashing F and comparing the resulting hash to the one stored in readonly memory. If a mismatch is found, the system declares that file F is corrupt. The TripWire malware protection system [63] uses this mechanism to protect critical system files. What property should the hash function H satisfy for this integrity mechanism to be secure? Let F be a file protected by this system. Since the malware cannot alter the contents of the readonly storage, its only avenue for modifying F without being detected is to find another file F 0 such that H(F ) = H(F 0 ). Replacing F by F 0 would not be caught by this hashing system. However, finding such an F 0 will be difficult if H is collision resistant. Collision resistance, thus, implies that the malware cannot change F without being detected by the hash. This system stores all file hashes in readonly memory. When there are many files to protect the amount of readonly memory needed could become large. We can greatly reduce the size of readonly memory by viewing the entire set of file hashes as just another file stored on disk and denoted FH . We store the hash of FH in readonly memory, as described in Fig. 8.2. Then readonly memory contains a single hash value. To verify file integrity of some file F we first verify integrity of the file FH by hashing the contents of FH and comparing the result to the value in readonly memory. Then we verify integrity of F by hashing F and comparing the result with the corresponding hash stored in FH . We describe a more efficient solution using authentication trees in Section 8.9. In the introduction to Chapter 6 we proposed a MACbased file integrity system. The system stored a tag of every file along with the file. We also needed a small amount of secret storage to store the user’s secret MAC key. This key was used every time file integrity was verified. In comparison, when using collision resistant hashing there are no secrets and there is no need for secret storage. Instead, we need a small amount of readonly storage for storing file hashes. Generally speaking, readonly storage is much easier to build than secret storage. Hence, collision resistance seems more appropriate for this particular application. In Chapter 13 we will develop an even better solution to this problem, using digital signatures, that does not need readonly storage or online secret storage. Security without collision resistance. By extending the input to the hash function with a few random bits we can prove security for both applications above using a weaker notion of collision resistance called target collision resistance or TCR for short. We show in Section 8.11.2 how to use TCR for both file integrity and for extending cryptographic primitives. The downside is that the 281
resulting tags are longer than the ones obtained from collision resistant hashing. Hence, although in principle it is often possible to avoid relying on collision resistance, the resulting systems are not as efficient.
8.1
Definition of collision resistant hashing
A (keyless) hash function H : M ! T is an efficiently computable function from some (large) message space M into a (small) digest space T . We say that H is defined over (M, T ). We define collision resistance of H using the following (degenerate) game: Attack Game 8.1 (Collision Resistance). For a given hash function H over (M, T ) and adversary A, the adversary takes no input and outputs two messages m0 and m1 in M. We say that A wins the game if the pair m0 , m1 is a collision for H, namely m0 6= m1 and H(m0 ) = H(m1 ). We define A’s advantage with respect to H, denoted CRadv[A, H], as the probability that A wins the game. Adversary A is called a collision finder. 2 Definition 8.1. We say that a hash function H over (M, T ) is collision resistant if for all efficient adversaries A, the quantity CRadv[A, H] is negligible. At first glance, it may seem that collision resistant functions cannot exist. The problem is this: since M > T  there must exist inputs m0 and m1 in M that collide, namely H(m0 ) = H(m1 ). An adversary A that simply prints m0 and m1 and exits is an efficient adversary that breaks the collision resistance of H. We may not be able to write the explicit program code for A (since we do not know m0 , m1 ), but this A certainly exists. Consequently, for any hash function H defined over (M, T ) there exists some efficient adversary AH that breaks the collision resistance of H. Hence, it appears that no function H can satisfy Definition 8.1. The way out of this is that, formally speaking, our hash functions are parameterized by a system parameter: each choice of a system parameter describes a di↵erent function H, and so we cannot simply “hardwire” a fixed collision into an adversary: an e↵ective adversary must be able to efficiently compute a collision as a function of the system parameter. This is discussed in more depth in the Mathematical details section below.1
8.1.1
Mathematical details
As usual, we give a more mathematically precise definition of a collision resistant hash function using the terminology defined in Section 2.4. Definition 8.2 (Keyless hash functions). A (keyless) hash function is an efficient algorithm H, along with two families of spaces with system parameterization P : M = {M
,⇤ } ,⇤ ,
and
T = {T
,⇤ } ,⇤ ,
such that 1. M, and T are efficiently recognizable. 1
Some authors deal with this issue by have H take as input a randomly chosen key k, and giving k to the adversary at the beginning of this attack game. By viewing k as a system parameter, this approach is really the same as ours.
282
R
Adversary A
CRHF Challenger
⇤
P( ) ⇤
m0 , m1
Figure 8.3: Asymptotic version of Attack Game 8.1 2. Algorithm H is an efficient deterministic algorithm that on input and m 2 M ,⇤ , outputs an element of T ,⇤ .
2Z
1,
⇤ 2 Supp(P ( )),
In defining collision resistance we parameterize Attack Game 8.1 by the security parameter . The asymptotic game is shown in Fig. 8.3. The advantage CRadv[A, H] is then a function of . Definition 8.1 should be read as saying that CRadv[A, H]( ) is a negligible function. It should be noted that the security and system parameters are artifacts of the formal framework that are needed to make sense of Definition 8.1. In the real world, however, these parameters are picked when the hash function is designed, and are ignored from that point onward. SHA256, for example, does not take either a security parameter or a system parameter as input.
8.2
Building a MAC for large messages
To exercise the definition of collision resistance, we begin with an easy application described in the introduction — extending the message space of a MAC. Suppose we are given a secure MAC I = (S, V ) for short messages. Our goal is to build a new secure MAC I 0 for much longer messages. We do so using a collision resistant hash function: I 0 computes a tag for a long message m by first hashing m to a short digest and then applying I to the digest, as shown in Fig. 8.1. More precisely, let H be a hash function that hashes long messages in M to short digests in TH . Suppose I is defined over (K, TH , T ). Define I 0 = (S 0 , V 0 ) for long messages as follows: S 0 (k, m) := S(k, H(m) )
and
V 0 (k, m) := V (k, H(m) )
(8.1)
Then I 0 authenticates long messages in M. The following easy theorem shows that I 0 is secure, assuming H is collision resistant. Theorem 8.1. Suppose the MAC system I is a secure MAC and the hash function H is collision resistant. Then the derived MAC system I 0 = (S 0 , V 0 ) defined in (8.1) is a secure MAC. In particular, suppose A is a MAC adversary attacking I 0 (as in Attack Game 6.1). Then there exist a MAC adversary BI and an efficient collision finder BH , which are elementary wrappers around A, such that MACadv[A, I 0 ] MACadv[BI , I] + CRadv[BH , H].
283
It is clear that collision resistance of H is essential for the security of I 0 . Indeed, if an adversary can find a collision m0 , m1 on H, then he can win the MAC attack game as follows: submit m0 to the MAC challenger for signing, obtaining a tag t0 := S(k, H(m0 )), and then output the messagetag pair (m1 , t0 ). Since H(m0 ) = H(m1 ), the tag t0 must be a valid tag on the message m1 . Proof idea. Our goal is to show that no efficient adversary can win the MAC Attack Game 6.1 for our new MAC system I 0 . An adversary A in this game asks the challenger to MAC a few long messages m1 , m2 , . . . 2 M and then tries to invent a new valid messageMAC pair (m, t). If A is able to produce a valid forgery (m, t) then one of two things must happen: 1. either m collides with some query mi from A, so that H(m) = H(mi ) and m 6= mi ; 2. or m does not collide under H with any of A’s queries m1 , m2 , . . . 2 M. It should be intuitively clear that if A produces forgeries of the first type then A can be used to break the collision resistance of H since m and mi are a valid collision for H. On the other hand, if A produces forgeries of the second type then A can be used to break the MAC system I: the pair (H(m), t) is a valid MAC forgery for I. Thus, if A wins the MAC attack game for I 0 we break one of our assumptions. 2 Proof. We make this intuition rigorous. Let m1 , m2 , . . . 2 M be A’s queries during the MAC attack game and let (m, t) 2 M ⇥ T be the adversary’s output, which we assume is not among the signed pairs. We define three events: • Let X be the event that adversary A wins the MAC Attack Game 6.1 with respect to I 0 . • Let Y denote the event that some mi collides with m under H, that is, for some i we have H(m) = H(mi ) and m 6= mi . • Let Z denote the event that A wins Attack Game 6.1 on I 0 and event Y did not occur. Using events Y and Z we can rewrite A’s advantage in winning Attack Game 6.1 as follows: MACadv[A, I 0 ] = Pr[X] Pr[X ^ ¬Y ] + Pr[Y ] = Pr[Z] + Pr[Y ] To prove the theorem we construct a collision finder BH and a MAC adversary BI such that Pr[Y ] = CRadv[BH , H]
and
Pr[Z] = MACadv[BI , I].
Both adversaries are straightforward. Adversary BH plays the role of challenger to A in the MAC attack game, as follows: Initialization: k R K Upon receiving a signing query mi 2 M from A do: ti R S(k, H(mi ) ) Send ti to A Upon receiving the final messagetag pair (m, t) from A do: if H(m) = H(mi ) and m 6= mi for some i then output the pair (m, mi )
284
(8.2)
MAC Adversary BI attacking I Adversary A
MAC Challenger hi
hi
H(mi )
ti 2 T
mi 2 M ti 2 T
(H(m), t)
(m, t)
Figure 8.4: Adversary BI in the proof of Theorem 8.1 Algorithm BH responds to A’s signature queries exactly as in a real MAC attack game. Therefore, event Y happens during the interaction with BH with the same probability that it happens in a real MAC attack game. Clearly when event Y happens, AH succeeds in finding a collision for H. Hence, CRadv[BH , H] = Pr[Y ] as required. MAC adversary BI is just as simple and is shown in Fig. 8.4. When A outputs the final messagetag pair (m, t) adversary BI outputs (H(m), t). When event Z happens we know that V 0 (k, m, t) outputs accept and the pair (m, t) is not equal to any of (m1 , t1 ), (m2 , t2 ), . . . 2 M ⇥ T . Furthermore, since event Y does not happen, we know that (H(m), t) is not equal to any of (H(m1 ), t1 ), (H(m2 ), t2 ), . . . 2 TH ⇥ T . It follows that (H(m), t) is a valid existential forgery for I. Hence, BI succeeds in creating an existential forgery with the same probability that event Z happens. In other words, MACadv[BI , I] = Pr[Z], as required. The proof now follows from (8.2). 2
8.3
Birthday attacks on collision resistant hash functions
Cryptographic hash functions are most useful when the output digest size is small. The challenge is to design hash functions whose output is as short as possible and yet finding collisions is difficult. It should be intuitively clear that the shorter the digest, the easier it is for an attacker to find collisions. To illustrate this, consider a hash function H that outputs `bit digests for some small `. Clearly, by hashing 2` + 1 distinct messages the attacker will find two messages that hash to the same digest and will thus break collision resistance of H. This bruteforce attack will break the collision resistance of any hash function. Hence, for instance, hash functions that output 16bit digests cannot be collision resistant — a collision can always be found using only 216 + 1 = 65537 evaluations of the hash. Birthday attacks. A far more devastating attack can be built using the birthday paradox discussed in Section B.1 in the appendix. Let H be a hash function defined over (M, T ) and set N := T . For standard hash functions N is quite large, for example N = 2256 for SHA256. Throughout this section we will assume that the size of M is at least 100N . This basically means that messages being hashed are slightly longer than the output digest. We describe a general colli
285
p sion finder that finds collisions for H after an expected O( N ) evaluations of H. For comparison, the bruteforce attack above took O(N ) evaluations. This more efficient collision finder forces us to use much larger digests. p The birthday collision finder for H works as follows: it chooses s ⇡ N random and independent messages, m1 , . . . , ms R M, and looks for a collision among these s messages. We will show that the birthday paradox implies that a collision is likely to exist among these messages. More precisely, the birthday collision finder works as follows: Algorithm BirthdayAttack: p 1. Set s d2 N e + 1 2. Generate s uniform random messages m1 , . . . , ms in M 3. Compute xi H(mi ) for all i = 1, . . . , s 4. Look for distinct i, j 2 {1, . . . , s} such that H(mi ) = H(mj ) 5. If such i, j exist and mi 6= mj then 6. output the pair (mi , mj ) l p m We argue that when the adversary picks s := 2 N + 1 random messages in M, then with probability at least 1/2, there will exist distinct i, j such that H(mi ) = H(mj ) and mi 6= mj . This means that the algorithm will output a collision with probability at least 1/2. Lemma 8.2. Let m1 , . . . , ms be the random messages sampled in Step 2. Assume M 100N . Then with probability at least 1/2 there exists i, j in {1, . . . , s} such that H(mi ) = H(mj ) and mi 6= mj . Proof. For i = 1, . . . , s let xi := H(mi ). First, we argue that two of the xi values will collide with probability at least 3/4. If the xi were uniformly distributed in T then this would follow immediately from part (i) of Theorem B.1. Indeed, if the xi were independent and uniform in T a collision among the xi will occur with probability at least 1 e s(s 1)/2N 1 e 2 3/4. However, in reality, the function H(·) might bias the output distribution. Even though the mi are sampled uniformly from M, the resulting xi may not be uniform in T . As a simple example, consider a hash function H(·) that only outputs digests in a certain small subset of T . The resulting xi would certainly not be uniform in T . Fortunately (for the attacker) Corollary B.2 shows that nonuniform xi only increase the probability of collision. Since the xi are independent and identically distributed the corollary implies that a collision among the xi will occur with probability at least 1 e s(s 1)/2N 3/4 as required. Next, we argue that a collision among the xi is very likely to lead to a collision on H(·). Suppose xi = xj for some distinct i, j in {1, . . . , s}. Since xi = H(mi ) and xj = H(mj ), the pair mi , mj is a candidate for a collision on H(·). We just need to argue that mi 6= mj . We do so by arguing that all the m1 , . . . , ms are distinct with probability at least 4/5. This follows directly from part (ii) of Theorem B.1. Recall that M is greater than 100N . Since m1 , m2 , . . . are uniform and independent in M, and s < M/2, part (ii) of Theorem B.1 implies that the probability of collision among these mi is at most 1 e s(s 1)/100N 1/5. Therefore, the probability that no collision occurs is at least 4/5. In summary, for the algorithm to discover a collision for H(·) it is sufficient that both a collision occurs on the xi values and no collision occurs on the mi values. This happens with probability at least 3/4 1/5 > 1/2, as required. 2
286
p Variations. Algorithm BirthdayAttack requires O( N ) memory space, which can be quite large: larger than the size of commercially available disk farms. However, a p modified birthday collision finder, described in Exercise 8.7, will find a collision with an expected 4 N evaluations of the hash function and constant memory space. p The birthday p attack is likely to fail if one makes fewer than N queries to H(·). Suppose we only make s = ✏ N queries to H(·), for some small ✏ 2 [0, 1]. For simplicity we assume that H(·) outputs digests distributed uniformly in T . Then part (ii) of Theorem B.1 shows that the 2 probability of finding a collision degrades exponentially to approximately 1 e (✏ ) ⇡ ✏2 . Put di↵erently, if after evaluating the hash function s times an adversary should obtain a collision with probability at most , then we need the digest space T to satisfy T  s2 / . For 80 example, if after 2 evaluations of H a collision should be found with probability at most 2 80 then the digest size must be at least 240 bits. Cryptographic hash functions such as SHA256 output a 256bit digest. Other hash functions, such as SHA384 and SHA512, output even longer digests, namely, 384 and 512 bits respectively.
8.4
The MerkleDamg˚ ard paradigm
We now turn to constructing collision resistant hash functions. Many practical constructions follow the MerkleDamg˚ ard paradigm: start from a collision resistant hash function that hashes short messages and build from it a collision resistant hash function that hashes much longer messages. This paradigm reduces the problem of constructing collision resistant hashing to the problem of constructing collision resistance for short messages, which we address in the next section. Let h : X ⇥ Y ! X be a hash function. We shall assume that Y is of the form {0, 1}` for some ard `. While it is not necessary, typically X is of the form {0, 1}n for some n. The MerkleDamg˚ function derived from h, denoted HMD and shown in Fig. 8.5, is a hash function defined over ({0, 1}L , X ) that works as follows (the pad PB is defined below): input: M 2 {0, 1}L output: a tag in X ˆ M M k PB // pad with PB to ensure that the length of M is a multiple of ` bits ˆ into consecutive `bit blocks so that partition M ˆ = m1 k m2 k · · · k ms where m1 , . . . , ms 2 {0, 1}` M t0 IV 2 X for i = 1 to s do: ti h(ti 1 , mi ) output ts The function SHA256 is a MerkleDamg˚ ard function where ` = 512 and n = 256. Before proving collision resistance of HMD let us first introduce some terminology for the various elements in Fig. 8.5: • The hash function h is called the compression function of H. • The constant IV is called the initial value and is fixed to some prespecified value. One could take IV = 0n , but usually the IV is set to some complicated string. For example, SHA256
287
m1
t0 := IV
m2
h
t1
ms
···
h
ts
t2
PB
1
h
ts := H(M )
Figure 8.5: The MerkleDamg˚ ard iterated hash function uses a 256bit IV whose value in hex is IV := 6A09E667 BB67AE85 3C6EF372 A54FF53A 510E527F 9B05688C 1F83D9AB 5BE0CD19. • The variables m1 , . . . , ms are called message blocks. • The variables t0 , t1 , . . . , ts 2 X are called chaining variables. • The string PB is called the padding block. It is appended to the message to ensure that the message length is a multiple of ` bits. The padding block PB must contain an encoding of the input message length. We will use this in the proof of security below. A standard format for PB is as follows: PB := 100 . . . 00 k hsi where hsi is a fixedlength bit string that encodes, in binary, the number of `bit blocks in M . Typically this field is 64bits which means that messages to be hashed are less than 264 blocks long. The ‘100 . . . 00’ string is a variable length pad used to ensure that the total message length, including PB, is a multiple of `. The variable length string ‘100 . . . 00’ starts with a ‘1’ to identify the position where the pad ends and the message begins. If the message length is such that there is no space for PB in the last block (for example, if the message length happens to be a multiple of `), then an additional block is added just for the padding block. Security of MerkleDamg˚ ard. Next we prove that the MerkleDamg˚ ard function is collision resistant, assuming the compression function is. Theorem 8.3 (MerkleDamg˚ ard). Let L be a polybounded length parameter and let h be a collision resistant hash function defined over (X ⇥ Y, X ). Then the MerkleDamg˚ ard hash function HMD derived from h, defined over ({0, 1}L , X ), is collision resistant. In particular, for every collision finder A attacking HMD (as in Attack Game 8.1) there exists a collision finder B attacking h, where B is an elementary wrapper around A, such that CRadv[A, HMD ] = CRadv[B, h].
Proof. The collision finder B for finding hcollisions works as follows: it first runs A to obtain two distinct messages M and M 0 in {0, 1}L such that HMD (M ) = HMD (M 0 ). We show that B can use 288
M and M 0 to find an hcollision. To do so, B scans M and M 0 starting from the last block and works its way backwards. To simplify the notation, we assume that M and M 0 already contain the appropriate padding block PB in their last block. Let M = m1 m2 . . . mu be the u blocks of M and let M 0 = m01 m02 . . . m0v be the v blocks of M 0 . We let t0 , t1 , . . . , tu 2 X be the chaining values for M and t00 , t01 , . . . , t0s 2 X be the chaining values for M 0 . The very last application of h gives the final output digest and since HMD (M ) = HMD (M 0 ) we know that h(tu 1 , mu ) = h(t0v 1 , m0v ). If either tu 1 6= t0v 1 or mu 6= m0v then the pair of inputs (tu 1 , mu ) and (t0v 1 , m0v ) is an hcollision. B outputs this collision and terminates. Otherwise, tu 1 = t0v 1 and mu = m0v . Recall that the padding blocks are contained in mu and 0 mv and these padding blocks contain an encoding of u and v. Therefore, since mu = m0v we deduce that u = v so that M and M 0 must contain the same number of blocks. At this point we know that u = v, mu = m0u , and tu 1 = t0u 1 . We now consider the secondtolast block. Since tu 1 = t0u 1 we know that h(tu
2 , mu 1 )
= h(t0u
0 2 , mu 1 ).
As before, if either tu 2 6= t0u 2 or mu 1 6= m0u 1 then B just found an hcollision. It outputs this collision and terminates. Otherwise, we know that tu 2 = t0u 2 and mu 1 = m0u 1 and mu = m0u . We now consider the third block from the end. As before, we either find an hcollision or deduce that mu 2 = m0u 2 and tu 3 = t0u 3 . We keep iterating this process moving from right to left one block at a time. At the ith block one of two things happens. Either the pair of messages (ti 1 , mi ) and (t0i 1 , m0i ) is an hcollision, in which case B outputs this collision and terminates. Or we deduce that ti 1 = t0i 1 and mj = m0j for all j = i, i + 1, . . . , u. Suppose this process continues all the way to the first block and we still did not find an hcollision. Then at this point we know that mi = m0i for i = 1, . . . , u. But this implies that M = M 0 contradicting the fact that M and M 0 were a collision for HMD . Hence, since M 6= M 0 , the process of scanning blocks of M and M 0 from right to left must produce an hcollision. We conclude that B breaks the collision resistance of h as required. In summary, we showed that whenever A outputs an HMD collision, B outputs an hcollision. Hence, CRadv[A, HMD ] = CRadv[B, h] as required. 2 Variations. Note that the MerkleDamg˚ ard construction is inherently sequential — the ith block cannot be hashed before hashing all previous blocks. This makes it difficult to take advantage of hardware parallelism when available. In Exercise 8.8 we investigate a di↵erent hash construction that is better suited for a multiprocessor machine. The MerkleDamg˚ ard theorem (Theorem 8.3) shows that collision resistance of the compression function is sufficient to ensure collision resistance of the iterated function. This condition, however, is not necessary. Black, Rogaway, and Shrimpton [17] give several examples of compression functions that are clearly not collision resistant, and yet the resulting iterated MerkleDamg˚ ard functions are collision resistant.
289
8.4.1
Joux’s attack
We briefly describe a cute attack that applies specifically to MerkleDamg˚ ard hash functions. Let ard hash functions that output tags in X := {0, 1}n . Define H12 (M ) := H1 and H2 be MerkleDamg˚ 2n H1 (M ) k H2 (M ) 2 {0, 1} . One would expect that finding a collision for H12 should take time at least ⌦(2n ). Indeed, this would be the case if H1 and H2 were independent random functions. ard functions we can find collisions for H in We show that when H1 and H2 are MerkleDamg˚ time approximately n2n/2 which is far less than 2n . This attack illustrates that our intuition about random functions may lead to incorrect conclusions when applied to a MerkleDamg˚ ard function. We say that an scollision for a hash function H is a set of messages M1 , . . . , Ms 2 M such that ard function in H(M1 ) = . . . = H(Ms ). Joux showed how to find an scollision for a MerkleDamg˚ 1/2 n/2 time O((log2 s)X  ). Using Joux’s method we can find a 2 collision M1 , . . . , M2n/2 for H1 in time O(n2n/2 ). Then, by the birthday paradox it is likely that two of these messages, say Mi , Mj , are also a collision for H2 . This pair Mi , Mj is a collision for both H1 and H2 and therefore a collision for H12 . It was found in time O(n2n/2 ), as promised. Finding scollisions. To find an scollision, let H be a MerkleDamg˚ ard function over (M, X ) built from a compression function h. We find an scollision M1 , . . . , Ms 2 M where each message Mi contains log2 s blocks. For simplicity, assume that s is a power of 2 so that log2 s is an integer. ard construction. As usual, we let t0 denote the Initial Value (IV) used in the MerkleDamg˚ The plan is to use the birthday attack log2 s times on the compression function h. We first spend time 2n/2 to find two distinct blocks m0 , m00 such that (t0 , m0 ) and (t0 , m00 ) collide under h. Let t1 := h(t0 , m0 ). Next we spend another 2n/2 time to find two distinct blocks m1 , m01 such that (t1 , m1 ) and (t1 , m01 ) collide under h. Again, we let t2 := h(t1 , m1 ) and repeat. We iterate this process b := log2 s times until we have b pairs of blocks: (mi , m0i )
for i = 0, 1, . . . b
1
that satisfy
h(ti , mi ) = h(ti , m0i ).
Now, consider the message M = m0 m1 . . . mb 1 . The main point is that replacing any block mi in this message by m0i will not change the chaining value ti+1 and therefore the value of H(M ) will not change. Consequently, we can replace any subset of m0 , . . . , mb 1 by the corresponding blocks in m00 , . . . , m0b 1 without changing H(M ). As a result we obtain s = 2b messages m0 m1 . . . mb m00 m1 . . . mb m0 m01 . . . mb m00 m01 . . . mb .. . m00 m01 . . . m0b
1 1 1 1
1
that all hash to same value under H. In summary, we found a 2b collision in time O(b2n/2 ). As explained above, this lets us find collisions for H(M ) := H1 (M ) k H2 (M ) in time O(n2n/2 ).
8.5
Building Compression Functions
The MerkleDamg˚ ard paradigm shows that to construct a collision resistant hash function for long messages it suffices to construct a collision resistant compression function h for short blocks. In 290
y := mi 2 K
x := ti
1
E
L
ti := E(mi , ti
1)
ti
1
2X
Figure 8.6: The DaviesMeyer compression function this section we describe a few candidate compression functions. These constructions fall into two categories: • Compression functions built from a block cipher. The most widely used method is called DaviesMeyer. The SHA family of cryptographic hash functions all use DaviesMeyer. • Compression functions using number theoretic primitives. These are elegant constructions with clean proofs of security. Unfortunately, they are generally far less efficient than the first method.
8.5.1
A simple but inefficient compression function
We start with a compression function built using modular arithmetic. Let p be a large prime such that q := (p 1)/2 is also prime. Let x and y be suitably chosen integers in the range [1, q]. Consider the following simple compression function that takes as input two integers in [1, q] and outputs an integer in [1, q]: ( z if z q, H(a, b) = abs(xa y b mod p), where abs(z) := (8.3) p z if z > q. We will show later in Exercise 10.18 that this function is collision resistant assuming a certain standard number theoretic problem is hard. Applying the MerkleDamg˚ ard paradigm to this function gives a collision resistant hash function for arbitrary size inputs. Although this is an elegant collision resistant hash with a clean security proof, it is far less efficient than functions derived from the DaviesMeyer construction and, as a result, is hardly ever used in practice.
8.5.2
DaviesMeyer compression functions
In Chapter 4 we spent the e↵ort to build secure block ciphers like AES. It is natural to ask whether we can leverage these constructions to build fast compression functions. The DaviesMeyer method enables us to do just that, but security can only be shown in the ideal cipher model. Let E = (E, D) be a block cipher over (K, X ) where X = {0, 1}n . The DaviesMeyer compression function derived from E maps inputs in X ⇥ K to outputs in X . The function is defined as follows: hDM (x, y) := E(y, x) x and is illustrated in Fig. 8.6. In symbols, hDM is defined over (X ⇥ K, X ). 291
MatyasMeyerOseas
MiyaguchiPreneel y := mi 2 X
y := mi 2 X
x := ti
1
g
E
L
x := ti
ti 2 X
1
g
E
L
ti 2 X
Figure 8.7: Other block cipher compression functions When plugging this compression function into the MerkleDamg˚ ard paradigm the inputs are a chaining variable x := ti 1 2 X and a message block y := mi 2 K. The output is the next chaining variable ti := E(mi , ti 1 ) ti 1 2 X . Note that the message block is used as the block cipher key which seems a bit odd since the adversary has full control over the message. Nevertheless, we will ard function is collision show that hDM is collision resistant and therefore the resulting MerkleDamg˚ resistant. When using hDM in MerkleDamg˚ ard the block cipher key (mi ) changes from one message block to the next, which is an unusual way of using a block cipher. Common block ciphers are optimized to encrypt long messages with a fixed key; changing the block cipher key on every block can slow down the cipher. Consequently, using DaviesMeyer with an o↵theshelf block cipher such as AES will result in a relatively slow hash function. Instead, one uses a custom block cipher specifically designed for rapid key changes. Another reason to not use an o↵theshelf block cipher in DaviesMeyer is that the block size may be too short, for example 128 bits for AES. An AESbased compression function would produce a 128bit output which is much too short for collision resistance: a collision could be found with only 264 evaluations of the function. In addition, o↵theshelf block ciphers use relatively short keys, say 128 bits long. This would result in MerkleDamg˚ ard processing only 128 message bits per round. Typical ciphers used in MerkleDamg˚ ard hash functions use longer keys (typically, 512bits or even 1024bits long) so that many more message bits are processed in every round. DaviesMeyer variants. The DaviesMeyer construction is not unique. Many other similar methods can convert a block cipher into a collision resistant compression function. For example, one could use MatyasMeyerOseas: h1 (x, y) := E(x, y) y MiyaguchiPreneel: h2 (x, y) := E(x, y) y x Or even: h3 (x, y) := E(x y, y) y or many other such variants. Preneel et al. [89] give twelve di↵erent variants that can be shown to be collision resistant. The MatyasMeyerOseas function h1 is similar to DaviesMeyer, but reverses the roles of the chaining variable and the message block — in h1 the chaining variable is used as the block cipher 292
key. The function h1 maps elements in (K ⇥ X ) to X . Therefore, to use h1 in MerkleDamg˚ ard we need an auxiliary encoding function g : X ! K that maps the chaining variable ti 1 2 X to an element in K, as shown in Fig. 8.7. The same is true for the MiyaguchiPreneel function h2 . The DaviesMeyer function does not need such an encoding function. We note that the MiyaguchiPreneel function has a minor security advantage over DaviesMeyer, as discussed in Exercise 8.14. Many other natural variants of DaviesMeyer are totally insecure. For example, for the following functions h4 (x, y) := E(y, x) y h5 (x, y) := E(x, x y)
x
we can find collisions in constant time (see Exercise 8.10).
8.5.3
Collision resistance of DaviesMeyer
We cannot prove that DaviesMeyer is collision resistant by assuming a standard complexity assumption about the block cipher. Simply assuming that E = (E, D) is a secure block cipher is insufficient for proving that hDM is collision resistant. Instead, we have to model the block cipher as an ideal cipher. We introduced the ideal cipher model back in Section 4.7. Recall that this is a heuristic technique in which we treat the block cipher as if it were a family of random permutations. If E = (E, D) is a block cipher with key space K and data block space X , then the family of random permutations is {⇧k }k 2K , where each ⇧k is a truly random permutation on X , and the ⇧k ’s collectively are mutually independent. Attack Game 8.1 can be adapted to the ideal cipher model, so that before the adversary outputs a collision, it may make a series of ⇧queries and ⇧ 1 queries to its challenger. • For a ⇧query, the adversary submits a pair (k , a ) 2 K ⇥ X , to which the challenger responds with b := ⇧k (a ). • For a ⇧ 1 query, the adversary submits a pair (k , b ) 2 K⇥X , to which the challenger responds with a := ⇧k 1 (b ). After making these queries, the adversary attempts to output a collision, which in the case of DaviesMeyer, means (x, y) 6= (x0 , y 0 ) such that ⇧y (x)
x = ⇧y0 (x0 )
x0 .
The adversary A’s advantage in finding a collision for hDM in the ideal cipher model is denoted CRic adv[A, hDM ], and security in the ideal cipher model means that this advantage is negligible for all efficient adversaries A. Theorem 8.4 (DaviesMeyer). Let hDM be the DaviesMeyer hash function derived from a block cipher E = (E, D) defined over (K, X ), where X  is large. Then hDM is collision resistant in the ideal cipher model. In particular, every collision finding adversary A that issues at most q idealcipher queries will satisfy CRic adv[A, hDM ] (q + 1)(q + 2)/X .
293
The theorem p shows that DaviesMeyer is an optimal compression function: the adversary must issue q = ⌦( X ) queries (and hence must run for at least that amount of time) if he is to find a collision for hDM with constant probability. No compression function can have higher security due to the birthday attack. Proof. Let A be a collision finder for hDM that makes at most a total of q ideal cipher queries. We shall assume that A is “reasonable”: before A outputs its collision attempt (x, y), (x0 , y 0 ), it makes corresponding ideal cipher queries: for (x, y), either a ⇧query on (y, x) or a ⇧ 1 query on (y, ·) that yields x, and similarly for (x0 , y 0 ). If A is not already reasonable, we can make it so by increasing total number of queries to at most q 0 := q + 2. So we will assume A is reasonable and makes at most q 0 ideal cipher queries from now on. For i = 1, . . . , q 0 , the ith ideal cipher query defines a triple (k i , a i , b i ): for a ⇧query (k i , a i ), we set b i := ⇧k i (a i ), and for a ⇧ 1 query (k i , b i ), we set a i := ⇧k 1 (b i ). We assume that A makes no i extraneous queries, so that no triples repeat. If the adversary outputs a collision, then by our reasonableness assumption, for some distinct pair of indices i, j = 1, . . . , q 0 , we have a i b i = a j b j . Let us call this event Z. So we have CRic adv[A, hDM ] Pr[Z]. Our goal is to show Pr[Z]
q 0 (q 0 1) , 2n
(8.4)
where X  = 2n . Consider any fixed indices i < j. Conditioned on any fixed values of the adversary’s coins and the first j 1 triples, one of a j and b j is completely fixed, while the other is uniformly distributed over a set of size at least X  j + 1. Therefore, Pr[a i
bi = aj
bj]
2n
1 . j+1
So by the union bound, we have 0
Pr[Z]
j 1 q X X
0
Pr[a i
bi = aj
j=1 i=1
bj]
q X j=1
j 2n
q0
X j 1 j+1 2n j=1
For q 0 2n 1 this bound simplifies to Pr[Z] q 0 (q 0 1)/2n . For q 0 > 2n Therefore, (8.4) holds for all q 0 . 2
8.6
1 q 0 (q 0 = q0 2(2n 1
1) . q0)
(8.5)
the bound holds trivially.
Case study: SHA256
The Secure Hash Algorithm (SHA) was published by NIST in 1993 [FIPS 180] as part of the design specification of the Digital Signature Standard (DSS). This hash function, often called SHA0, outputs 160bit digests. Two years later, in 1995, NIST updated the standard [FIPS 1801] by adding one extra instruction to the compression function. The resulting function is called SHA1. NIST gave no explanation for this change, but it was later found that this extra instruction is crucial for collision resistance. SHA1 became the defacto standard for collision resistant hashing and is very widely deployed. 294
Name SHA0 SHA1 SHA224 SHA256 SHA384 SHA512 MD4 MD5 Whirpool
year 1993 1995 2004 2002 2002 2002 1990 1992 2000
digest size 160 160 224 256 384 512 128 128 512
message block size 512 512 512 512 1024 1024 512 512 512
Speed2 MB/sec 153
best known attack time 239 263
111 99 255 57
21 230
Table 8.1: MerkleDamg˚ ard collision resistant hash functions The birthday attack can find collisions for SHA1 using an expected 280 evaluations of the function. In 2002 NIST added [FIPS 1802] two new hash functions to the SHA family: SHA256 and SHA512. They output larger digests (256 and 512bit digests respectively) and therefore provide better protection against the birthday attack. NIST also approved SHA224 and SHA384 which are obtained from SHA256 and SHA512 respectively by truncating the output to 224 and 384 bits. These and a few other proposed hash functions are summarized in Table 8.1. The years 2004–5 were bad years for collision resistant hash functions. A number of new attacks showed how to find collisions for a variety of hash functions. In particular, Wang, Yao, and Yao [103] presented a collision finder for SHA1 that uses 263 evaluations of the function — far less than the birthday attack. As a result SHA1 is no longer considered collision resistant. The current recommended practice is to use SHA256 which we describe here. The SHA256 function. SHA256 is a MerkleDamg˚ ard hash function using a DaviesMeyer compression function h. This h takes as input a 256bit chaining variable t and a 512bit message block m. It outputs a 256bit chaining variable. We first describe the SHA256 MerkleDamg˚ ard chain. Recall that the padding block PB in our description of MerkleDamg˚ ard contained a 64bit encoding of the number of blocks in the message being hashed. The same is true for SHA256 with the minor di↵erence that PB encodes the number of bits in the message. Hence, SHA256 can hash messages that are at most 264 1 bits long. The MerkleDamg˚ ard Initial Value (IV) in SHA256 is set to: IV := 6A09E667 BB67AE85 3C6EF372 A54FF53A 510E527F 9B05688C 1F83D9AB 5BE0CD19 2 {0, 1}256 written in base 16. Clearly the output of SHA256 can be truncated to obtain shorter digests at the cost of reduced security. This is, in fact, how the SHA224 hash function works — it is identical to SHA256 with two exceptions: (1) SHA224 uses a di↵erent initialization vector IV, and (2) SHA224 truncates the output of SHA256 to its left most 224 bits. 2
Performance numbers were provided by Wei Dai using the Crypto++ 5.6.0 benchmarks running on a 1.83 GhZ Intel Core 2 processor. Higher numbers are better.
295
Next, we describe the SHA256 DaviesMeyer compression function h. It is built from a block cipher which we denote by ESHA256 . However, instead of using XOR as in DaviesMeyer, SHA256 uses addition modulo 232 . That is, let x0 , x1 , . . . , x7 2 {0, 1}32
and
y0 , y1 , . . . , y7 2 {0, 1}32
x := x0 k · · · k x7 2 {0, 1}256
and
y := y0 k · · · k y7 2 {0, 1}256 .
and set Define: x y := (x0 + y0 ) k · · · k (x7 + y7 ) 2 {0, 1}256 Then the SHA256 compression function h is defined as: h(t, m) := ESHA256 (m, t)
t
where all additions are modulo 232 .
2 {0, 1}256 .
Our ideal cipher analysis of DaviesMeyer (Theorem 8.4) applies equally well to this modified function. The SHA256 block cipher. To complete the description of SHA256 it remains to describe the block cipher ESHA256 . The algorithm makes use of a few auxiliary functions defined in Table 8.2. Here, SHR and ROTR denote the standard shiftright and rotateright functions. The cipher ESHA256 takes as input a 512bit key k and a 256bit message t. We first break both the key and the message into 32bit words. That is, write: k := k0 k k1 k · · · k k15 2 {0, 1}512 t := t0 k t1 k · · · k t7 2 {0, 1}256
where each ki and ti is in {0, 1}32 . The code for ESHA256 is shown in Table 8.3. It iterates the same round function 64 times. In each round the cipher uses a round key Wi 2 {0, 1}32 defined recursively during the key setup step. One cipher round, shown in Fig. 8.8, looks like two adjoined Feistel rounds. The cipher uses 64 fixed constants K0 , K1 , . . . , K63 2 {0, 1}32 whose values are specified in the SHA256 standard. For example, K0 := 428A2F 98 and K1 := 71374491, written base 16. Interestingly, NIST never gave the block cipher ESHA256 an official name. The cipher was given the unofficial name SHACAL2 by Handschuh and Naccache (submission to NESSIE, 2000). Similarly, the block cipher underlying SHA1 is called SHACAL1. The SHACAL2 block cipher is identical to ESHA256 with the only di↵erence that it can encrypt using keys shorter than 512 bits. Given a key k 2 {0, 1}512 the SHACAL2 cipher appends zeros to the key to get a 512bit key. It then applies ESHA256 to the given 256bit message block. Decryption in SHACAL2 is similar to encryption. This cipher is well suited for applications where SHA256 is already implemented, thus reducing the overall size of the crypto code.
8.6.1
Other MerkleDamg˚ ard hash functions
MD4 and MD5. Two cryptographic hash functions designed by Rivest in 1990–1 [90, 91]. Both are MerkleDamg˚ ard hash functions that output a 128bit digest. They are quite similar, although MD5 uses a stronger compression function than MD4. Collisions for both hash functions can be found efficiently as described in Table 8.1. Consequently, these hash functions should no longer be used. 296
For x, y, z in {0, 1}32 define: SHRn (x) := (x >> n) ROTRn (x) := (x >> n) _ (x symbol, which indicates which input is to be viewed as a PRF key. Indeed, the reader will observe that we will treat the two evaluations of h that appear within the dotted boxes as evaluations of the PRF htop , so that the values labeled k10 and k20 in the figure are computed htop (k1 , IV) and k20 htop (k2 , IV). All of the other evaluations of h in the figure will be as k10 treated as evaluations of hbot . Our assumption will be that hbot and htop are both secure PRFs. Later, we will use the ideal cipher model to justify this assumption for the DaviesMeyer compression function (see Section 8.7.3). We will now sketch a proof of the following result: If hbot and htop are secure PRFs, then so is the twokey nest. The first observation is that the keys k1 and k2 are only used to derive k10 and k20 as k10 = htop (k1 , IV) and k20 = htop (k2 , IV). The assumption that htop is a secure PRF means that in the PRF attack game, we can e↵ectively replace k10 and k20 by truly random nbit strings. The resulting construction drawn in Fig. 8.10. All we have done here is to throw away all of the elements in Fig. 8.9 that are within the dotted boxes. The function in this new construction takes as input 300
m1
k10
···
ms k PBi
h
t
h
t k PBo
k20
h
Figure 8.10: A bitwise version of NMAC the two keys k10 and k20 and a message M . By the above observations, it suffices to prove that the construction in Fig. 8.10 is a secure PRF. Hopefully (without reading the caption), the reader will recognize the construction in Fig. 8.10 as none other than NMAC applied to hbot , which we introduced in Section 6.5.1 (in particular, take a look at Fig. 6.5b). Actually, the construction in Fig. 8.10 is a bitwise version of NMAC, obtained from the blockwise version via padding (as discussed in Section 6.8). Thus, security for the twokey nest now follows directly from the NMAC security theorem (Theorem 6.7) and the assumption that hbot is a secure PRF.
8.7.2
The HMAC standard
The HMAC standard is exactly the same as the twokey nest (Fig. 8.9), but with one important di↵erence: the keys k1 and k2 are not independent, but rather, are derived in a somewhat ad hoc way from a single key k. To describe this in more detail, we first observe that HMAC itself is somewhat byte oriented, so all strings are byte strings. Message blocks for the underlying MerkleDamg˚ ard hash are assumed to be B bytes (rather than ` bits). A key k for HMAC is a byte string of arbitrary length. To derive the keys k1 and k2 , which are byte strings of length B, we first make k exactly B bytes long: if the length of k is less than or equal to B, we pad it out with zero bytes; otherwise, we replace it with H(k) padded with zero bytes. Then we compute k1
k
ipad and k2
k
opad,
where ipad and opad (“i” and “o” stand for “inner” and “outer”) are Bbyte constant strings, defined as follows: ipad = the byte 0x36 repeated B times opad = the byte 0x5C repeated B times HMAC implemented using a hash function H is denoted HMACH. The most common HMACs used in practice are HMACSHA1 and HMACSHA256. The HMAC standard also allows the output 301
of HMAC to be truncated. For example, when truncating the output of SHA1 to 80 bits, the HMAC function is denoted HMACSHA180. Implementations of TLS 1.0, for example, are required to support HMACSHA196. Security of HMAC. Since the keys k10 , k20 are related — their XOR is equal to opad ipad — the security proof we gave for the twokey nest no longer applies: under the stated assumptions, we cannot justify the claim that the derived keys k10 , k20 are indistinguishable from random. One solution is to make a stronger assumption about the compression function h – one needs to assume that htop remains a PRF under a related key attack (as defined by Bellare and Kohno [6]). If h is itself a DaviesMeyer compression function, then this stronger assumption can be justified in the ideal cipher model.
8.7.3
DaviesMeyer is a secure PRF in the ideal cipher model
It remains to justify our assumption that the PRFs hbot and htop derived from h in (8.6) are secure. Suppose the compression function h is a DaviesMeyer function, that is h(x, y) := E(y, x) x for some block cipher E = (E, D). Then • hbot (k, m) := h(k, m) = E(m, k)
k
• htop (k, m) := h(m, k) = E(k, m)
m
is a PRF defined over(X , K, X ), and is a PRF defined over(K, X , X )
When E is a secure block cipher, the fact that htop is a secure PRF is trivial (see Exercise 4.1 part (c)). The fact that hbot is a secure PRF is a bit surprising — the message m given as input to hbot is used as the key for E. But m is chosen by the adversary and hence E is evaluated with a key that is completely under the control of the adversary. As a result, even though E is a secure block cipher, there is no security guarantee for hbot . Nevertheless, we can prove that hbot is a secure PRF, but this requires the ideal cipher model. Just assuming that E is a secure block cipher is insufficient. If necessary, the reader should review the basic concepts regarding the ideal cipher model, which was introduced in Section 4.7. We also used the ideal cipher model earlier in this chapter (see Section 8.5.3). In the ideal cipher model, we heuristically model a block cipher E = (E, D) defined over (K, X ) as a family of random permutations {⇧k }k 2K . We adapt the PRF Attack Game 4.2 to work in the ideal cipher model. The challenger, in addition to answering standard queries, also answers ⇧queries and ⇧ 1 queries: a ⇧query is a pair (k , a ) to which the challenger responds with b := ⇧k (a ); a ⇧ 1 query is a pair (k , b ) to which is the challenger responds with a := ⇧k 1 (b ). For a standard query m, the challenger responds with v := f (m): in Experiment 0 of the attack game, f is F (k, ·), where F is a PRF and k is a randomly chosen key; in Experiment 1, f is a truly random function. Moreover, in Experiment 0, F is evaluated using the random permutations in the role of E and D used in the construction of F . For our PRF hbot (k, m) = E(m, k) k = ⇧m (k) k. For an adversary A, we define PRFic adv[A, F ] to be the advantage in the modified PRF attack game, and security in the ideal cipher model means that this advantage is negligible for all efficient adversaries. Theorem 8.5 (Security of hbot ). Let E = (E, D) be a block cipher over (K, X ), where X  is large. Then hbot (k, m) := E(m, k) k is a secure PRF in the ideal cipher model. 302
In particular, for every PRF adversary A attacking hbot and making at most a total of Qic ideal cipher queries, we have 2Qic PRFic adv[A, hbot ] . X 
The bound in the theorem is fairly tight, as bruteforce key search gets very close to this bound. Proof. The proof will mirror the analysis of the EvanMansour/EX constructions (see Theorem 4.14 in Section 4.7.4), and in particular, will make use of the Domain Separation Lemma (see Theorem 4.15, also in Section 4.7.4). Let A be an adversary as in the statement of the theorem. Let pb be the probability that A outputs 1 in Experiment b of Attack Game 4.2, for b = 0, 1. So by definition we have PRFic adv[A, hbot ] = p0
p1 .
(8.7)
We shall prove the theorem using a sequence of two games, applying the Domain Separation Lemma. Game 0. The game will correspond to Experiment 0 of the PRF attack game in the idea cipher model. We can write the logic of the challenger as follows: Initialize: for each k 2 K, set ⇧k k R X
R
Perms[X ]
standard hbot query m: 1. c ⇧m (k) 2. v c k 3. return v The challenger in Game 0 processes ideal cipher queries exactly as in Game 0 of the proof of Theorem 4.14: ideal cipher ⇧query k , a : b ⇧k (a ) 1. 2. return b ideal cipher ⇧ 1 query k , b : a ⇧k 1 (b ) 1. 2. return a Let W0 be the event that A outputs 1 at the end of Game 0. It should be clear from construction that Pr[W0 ] = p0 . (8.8) Game 1. Just as in the proof of Theorem 4.14, we declare “by fiat” that standard queries and ideal cipher queries are processed using independent random permutations. In detail (changed from Game 0 are highlighted): 303
Initialize: for each k 2 K, set ⇧std,k k R X standard hbot query m: ⇧std,m (k) // 1. c 2. v c k 3. return v
R
Perms[X ] and ⇧ic,k
R
Perms[X ]
add k to sampled domain of ⇧std,m , add c to sampled range of ⇧std,m
The challenger in Game 1 processes ideal cipher queries exactly as in Game 1 of the proof of Theorem 4.14: ideal cipher ⇧query k , a : b ⇧ic,k (a ) // add a to sampled domain of ⇧ic,k , add b to sampled range of ⇧ic,k 1. 2. return b ideal cipher ⇧
1 query
⇧ic,1k (b )
1.
a
2.
return a
//
k , b: add a to sampled domain of ⇧ic,k , add b to sampled range of ⇧ic,k
Let W1 be the event that A outputs 1 at the end of Game 1. Consider an input/output pair (m, v) for a standard query in Game 2. Observe that k is the only item ever added to the sampled domain of ⇧std,m (k), and c = v k is the only item ever added to the sampled range of ⇧std,m (k). In particular, c is generated at random and k remains perfectly hidden (i.e., is independent of the adversary’s view). Thus, from the adversary’s point of view, the standard queries behave identically to a random function, and the ideal cipher queries behave like ideal cipher queries for an independent ideal cipher. In particular, we have Pr[W1 ] = p1 . (8.9) Finally, we use the Domain Separation Lemma to analyze Pr[W0 ] Pr[W1 ]. The domain separation failure event Z is the event that in Game 1, the sampled domain of one of the ⇧std,m ’s overlaps with the sampled domain of one of the ⇧ic,k ’s, or the sampled range of one of the ⇧std,m ’s overlaps with the sampled range of one of the ⇧ic,k ’s. The Domain Separation Lemma tells us that Pr[W0 ]
Pr[W1 ] Pr[Z].
(8.10)
If Z occurs, then for some input/output triple (k , a , b ) corresponding to an ideal cipher query, k = m was the input to a standard query with output v, and either (i) a = k, or (ii) b = v
k.
For any fixed triple (k , a , b ), by the independence of k, conditions (i) and (ii) each hold with probability 1/X , and so by the union bound Pr[Z] The theorem now follows from (8.7)–(8.11). 2 304
2Qic . X 
(8.11)
8.8
The Sponge Construction and SHA3
For many years, essentially all collision resistant hash functions were based on the MerkleDamg˚ ard paradigm. Recently, however, an alternative paradigm has emerged, called the sponge construction. Like MerkleDamg˚ ard, it is a simple iterative construction built from a more primitive function; however, instead of a compression function h : {0, 1}n+` ! {0, 1}n , a permutation ⇡ : {0, 1}n ! {0, 1}n is used. We stress that unlike a block cipher, the function ⇡ has no key. There are two other highlevel di↵erences between the sponge and MerkleDamg˚ ard that we should point out: • On the negative side, it is not known how to reduce the collision resistance of the sponge to a concrete security property of ⇡. The only known analysis of the sponge is in the ideal permutation model, where we (heuristically) model ⇡ as a truly random permutation ⇧. • On the positive side, the sponge is designed to be used flexibly and securely in a variety of applications where collision resistance is not the main property we need. For example, in Section 8.7, we looked at several possible ways to convert a hash function H into a PRF F . We saw, in particular, that the intuitive idea of simply prepending the key, defining ard hash. Fpre (k, M ) := H(k k M ), does not work when H instantiated with a MerkleDamg˚ The sponge avoids these problems: it allows one to hash variable length inputs to variable length outputs, and if we model ⇡ as a random permutation, then one can argue that for all intents and purposes, the sponge is a random function (we will discuss this in more detail in Section 8.10). In particular, the construction Fpre is secure when H is instantiated with a sponge hash. A new hash standard, called SHA3, is based on the sponge construction. After giving a description and analysis of the general sponge construction, we discuss some of the particulars of SHA3.
8.8.1
The sponge construction
We now describe the sponge construction. In addition specifying a permutation ⇡ : {0, 1}n ! {0, 1}n , we need to specify two positive integers numbers r and c such that n = r + c. The number r is called the rate of the sponge: larger rate values lead to faster evaluation. The number c is called the capacity of the sponge: larger capacity values lead to better security bounds. Thus, di↵erent choices of r and c lead to di↵erent speed/security tradeo↵s. The sponge allows variable length inputs. To hash a long message M 2 {0, 1}L , we first append a padding string to M to make its length a multiple of r, and then break the padded M into a sequence of rbit blocks m1 , . . . , ms . The requirements of the padding procedure are minimal: it just needs to be injective. Just adding a string of the form 10⇤ suffices, although in SHA3 a pad of the form 10⇤ 1 is used: this latter padding has the e↵ect of encoding the rate in the last block and helps to analyze security in applications that use the same sponge with di↵erent rates; however, we will not explore these use cases here. Note that an entire dummy block may need to be added if the length of M is already at or near a multiple of r. The sponge allows variable length outputs. So in addition to a message M 2 {0, 1}L as above, it takes as input a positive integer v, which specifies the number of output bits. Here is how the sponge works: 305
Figure 8.11: The sponge construction Input: M 2 {0, 1}L and ` > 0 Output: a tag h 2 {0, 1}v // Absorbing stage Pad M and break into rbit blocks m1 , . . . , ms h 0n for i 1 to s do mi k 0c 2 {0, 1}n m0i h ⇡(h m0i ) // Squeezing stage z h[0 . . r 1] for i 1 to dv/re do h ⇡(h) z z k (h[0 . . r output z[0 . . v 1]
1])
The diagram in Fig. 8.11 may help to clarify the algorithm. The sponge runs in two stages: the “absorbing stage” where the message blocks get “mixed in” to a chaining variable h, and a “squeezing stage” where the output is “pulled out” of the chaining variable. Note that input blocks and output blocks are rbit strings, so that the remaining c bits of the chaining variable cannot be directly tampered with or seen by an attacker. This is what gives the sponge its security, and is the reason why c must be large. Indeed, if the sponge has small capacity, it is easy to find collisions (see Exercise 8.20). In the SHA3 standard, the sponge construction is intended to be used as a collision resistant hash, and the output length is fixed to a value v r, and so the squeezing stage simply outputs the first v bits of the output h of the absorbing stage. We will now prove that this version of the sponge is collision resistant in the ideal permutation model, assuming 2c and 2v are both superpoly. Theorem 8.6. Let H be the hash function obtained from a permutation ⇡ : {0, 1}n ! {0, 1}n , with capacity c, rate r (so n = r + c), and output length v r. In the ideal permutation model, where 306
⇡ is modeled as a random permutation ⇧, the hash function H is collision resistant, assuming 2v and 2c are superpoly. In particular, for every collision finding adversary A, if the number of idealpermutation queries plus the number of rbit blocks in the output messages of A is bounded by q, then CRic adv[A, H]
q(q 1) q(q + 1) + . 2v 2c
Proof. As in the proof of Theorem 8.4, we assume our collisionfinding adversary is “reasonable”, in the sense that it makes ideal permutation queries corresponding to its output. We can easily convert an arbitrary adversary into a reasonable one by forcing the adversary evaluate the hash function on its output messages if it has not done so already. As we have defined it, q will be an upper bound on the total number of ideal permutation queries made by our reasonable adversary. So from now on, we assume a reasonable adversary A that makes at most q queries, and we bound the probability that such A finds anything during its queries that can be “assembled” into a collision (we make this more precise below). We also assume that no queries are redundant. This means that if the adversary makes a ⇧query on a yielding b = ⇧(a ), then the adversary never makes a ⇧ 1 query on b , and never makes another ⇧query on a ; similarly, if the adversary makes a ⇧ 1 query on b yielding a = ⇧ 1 (b ), then the adversary never makes a ⇧query on a , and never makes another ⇧ 1 query on b . Of course, there is no need for the adversary to make such redundant queries, which is why we exclude them; moreover, doing so greatly simplifies the “bookkeeping” in the proof. It helps to visualize the adversary’s attack as building up a directed graph G. The nodes in G consist of the set of all 2n bit strings of length n. The graph G starts out with no edges, and every query that A makes adds an edge to the graph: an edge a ! b is added if A makes a ⇧query on a that yields b or a ⇧ 1 query on b that yields a . Notice that if we have an edge a ! b , then ⇧(a ) = b , regardless of whether that edge was added via a ⇧query or a ⇧ 1 query. We say that an edge added via a ⇧query is a forward edge, and one added via a ⇧ 1 query is a back edge. Note that the assumption that the adversary makes no redundant queries means that an edge gets added only once to the graph, and its classification is uniquely determined by the type of query that added the edge. We next define a notion of special type of path in the graph that corresponds to sponge evaluation. For an nbit string z , let R(z ) be the first r bits of z and C(z ) be the last c bits of z . We refer to R(z ) as the Rpart of z and C(z ) as the Cpart of z . For s 1, a Cpath of length s is a sequence of 2s nodes a 0, b 1, a 1, b 2, a 2, . . . , b s 1, a s 1, b s, where • C(a 0 ) = 0c and for i = 1, . . . , s • G contains edges a i
1
1, we have C(b i ) = C(a i ), and
! b i for i = 1, . . . , s.
For such a path p, the message of p is defined as (m0 , . . . , ms m0 := R(a 0 )
and
mi := R(b i )
307
1 ),
where
R(a i ) for i = 1, . . . , s
1.
and the result of p is defined to be ms := R(b s ). Such a Cpath p corresponds to evaluating the sponge at the message (m0 , . . . , ms 1 ) and obtaining the (untruncated) output ms . Let us write such a path as m0 a 0 ! b 1 m1 a 1 ! · · · ! b s
2 ms 2 a s 2
! bs
1 ms 1 a s 1
! b s ms .
(8.12)
The following diagram illustrates a Cpath of length 3.
a0
!
m0 = R(a 0 ) 0c = C(a 0 )
b1 a1
m1 = R(b 1 )
R(a 1 )
C(b 1 ) = C(a 1 )
!
b2 a2
m2 = R(b 2 )
!
R(a 2 )
b3
m3 = R(b 3 )
C(b 2 ) = C(a 2 )
The path has message (m0 , m1 , m2 ) and result m3 . Using the notation in (8.12), we write this path as m0 a 0 ! b 1 m1 a 1 ! b 2 m2 a 2 ! b 3 m3 . We can now state what a collision looks like in terms of the graph G. It is a pair of Cpaths on di↵erent messages but whose results agree on their first v bits (recall v r). Let us call such a pair of paths colliding. To analyze the probability of finding a pair of colliding paths, it will be convenient to define another notion. Let p and p0 be two Cpaths on di↵erent messages whose final edges are a s 1 ! b s and a 0t 1 ! b 0t . Let us call such a pair of paths problematic if (i) a s
1
= a 0t
1,
or
(ii) one of the edges in p or p0 are back edges. Let W be the event that A finds a pair of colliding paths. Let Z be the event that A finds a pair of problematic paths. Then we have Pr[W ] Pr[Z] + Pr[W and not Z].
(8.13)
First, we bound Pr[W and not Z]. For an nbit string z , let V (z ) be the first v bits of z , and we refer to V (z ) as the V part of z . Suppose A is able to find a pair of colliding paths that is not problematic. By definition, the final edges on these two paths correspond to ⇧queries on distinct inputs that yield outputs whose V parts agree. That is, if W and not Z occurs, then it must be the case that at some point A issued two ⇧queries on distinct inputs a and a 0 , yielding outputs b and b 0 such that V (b ) = V (b 0 ). We can use the union bound: for each pair of indices i < j, let Xij be the event that the ith query is a ⇧query on some value, say a , yielding b = ⇧(a ), and the jth query is also a ⇧query on some other value a 0 6= a , yielding b 0 = ⇧(a 0 ) such that V (b ) = V (b 0 ). If we fix i and j, fix the coins of A, and fix the outputs of all queries made prior to the jth query, then the values a , b , and a 0 are all fixed, but the value b 0 is uniformly distributed over a set of size at least 2n j + 1. To get V (b ) = V (b 0 ), the value of b 0 must be equal to one of the 2n v strings whose first v bits agree with that of b , and so we have Pr[Xij ]
2n 2n
308
v
j+1
.
A simple calculation like that done in (8.5) in the proof of Theorem 8.4 yields Pr[W and not Z]
q(q 1) . 2v
(8.14)
Second, we bound Pr[Z], the probability that A finds a pair of problematic paths. The technical heart of the of the analysis is the following: Main Claim: If Z occurs, then one of the following occurs: (E1) some query yields an output whose Cpart is 0c , or (E2) two di↵erent queries yield outputs whose Cparts are equal. Just to be clear, (E1) means A made a query of the form: (i) a ⇧ 1 query on some value b such that C(⇧ value a such that C(⇧(a )) = 0c ,
1 (b ))
= 0c , or (ii) a ⇧ query on some
and (E2) means A made pair of queries of the form: (i) a ⇧query on some value a and a ⇧ 1 query on some value b , such that C(⇧(a )) = C(⇧ 1 (b )), or (ii) ⇧queries on two distinct values a and a 0 such that C(⇧(a )) = C(⇧(a 0 )). First, suppose A is able to find a problematic pair of paths, and one of the paths contain a back edge. So at the end of the execution, there exists a Cpath containing one or more back edges. Let p be such a path of shortest length, and write it as in (8.12). We observe that the last edge in p is a back edge, and all other edges (if any) in p are forward edges. Indeed, if this is not the case, then we can delete this edge from p, obtaining a shorter Cpath containing a back edge, contradicting the assumption that p is a shortest path of this type. From this observation, we see that either: • s = 1 and (E1) occurs with the ⇧
1
query on b 1 , or
• s > 1 and (E2) occurs with the ⇧
1
query on b s and the ⇧query on a s
2.
Second, suppose A is able to find a problematic pair of paths, neither of which contains any back edges. Let us call these paths p and p0 . The argument in this case somewhat resembles the “backwards walk” in the MerkleDamg˚ ard analysis. Write p as in (8.12) and write p0 as m00 a 00 ! b 01 m01 a 01 ! · · · ! b 0t
0 0 2 mt 2 a t 2
! b 0t
0 0 1 mt 1 a t 1
! b 0t m0t .
We are assuming that (m0 , . . . , ms 1 ) 6= (m00 , . . . , m0t 1 ) but a s 1 = a 0t 1 , and that none of these edges are back edges. Let us also assume that we choose the paths so that they are shortest, in the sense that s + t is minimal among all Cpaths of this type. Also, let us assume that s t (swapping if necessary). There are a few cases: 1. s = 1 and t = 1. This case is impossible, since in this case the paths are just m0 a 0 ! b 1 m1 and m00 a 00 ! b 01 m01 , and we cannot have both m0 6= m00 and a 0 = a 00 . 2. s = 1 and t
2. In this case, we have a 0 = b 0t 309
1,
and so (E1) occurs on the ⇧query on a 0t
2.
3. s
2 and t
2. Consider the penultimate edges, which are forward edges:
as
2
! bs
1 ms 1 a s 1
a 0t
2
! b 0t
0 0 1 mt 1 a t 1 .
and We are assuming a s Rparts di↵er by ms
= a 0t 1 . Therefore, the Cparts of b s m0t 1 . There are two subcases: 1
1
1
and b 0t
1
are equal and their
= m0t 1 . We argue that this case is impossible. Indeed, in this case, we have b s 1 = b 0t 1 , and therefore a s 2 = a 0t 2 , while the truncated messages (m0 , . . . , ms 2 ) and (m01 , . . . , m0t 2 ) di↵er. Thus, we can simply throw away the last edge in each of the two paths, obtaining a shorter pair of paths that contradicts the minimality of s + t.
(a) ms
1
(b) ms 1 6= m0t 1 . In this case, we know: the Cparts of b s 1 and b 0t 1 are the same, but their Rparts di↵er, and therefore, a s 1 6= a 0t 2 . Thus, (E2) occurs on the ⇧queries on a s 2 and a 0t 2 . That proves the Main Claim. We can now turn to the problem of bounding the probability that either (E1) or (E2) occurs. This is really just the same type of calculation we did at least twice already, once above in obtaining (8.13), and earlier in the proof of Theorem 8.4. The only di↵erence from (8.13) is that we are now counting collisions on the Cparts, and we have a new type of “collision” to count, namely, “hitting 0c ” as in (E1). We leave it to the reader to verify: Pr[Z]
q(q + 1) . 2c
(8.15)
The theorem now follows from (8.13)–(8.15). 2
8.8.2
Case study: SHA3, SHAKE256, and SHAKE512
The NIST standard for SHA3 specifies a family of spongebased hash functions. At the heart of these hash functions is a permutation called Keccak, which maps 1600bit strings to 1600bit strings. We denote by Keccak[c] the sponge derived from Keccak with capacity c, and using the 10⇤ 1 padding rule. This is a function that takes two inputs: a message m and output length v. Here, the input m is an arbitrary bit string and the output of Keccak[c](m, v) is a vbit string. We will not describe the internal workings of the Keccak permutation; they can be found in the SHA3 standard. We just describe the di↵erent parameter choices that are standardized. The standard specifies four hash functions whose output lengths are fixed, and two hash functions with variable length outputs. Here are the four fixedlength output hash functions: • SHA3224(m) = Keccak[448](m k 01, 224); • SHA3256(m) = Keccak[512](m k 01, 256); • SHA3384(m) = Keccak[768](m k 01, 384); • SHA3512(m) = Keccak[1024](m k 01, 512). 310
Note the two extra padding bits that are appended to the message. Note that in each case, the capacity c is equal to twice the output length v. Thus, as the output length grows, the security provided by the capacity grows as well, and the rate — and, therefore, the hashing speed — decreases. Here are the two variablelength output hash functions: • SHAKE128(m, v) = Keccak[256](m k 1111, v); • SHAKE256(m, v) = Keccak[512](m k 1111, v). Note the four extra padding bits that are appended to the message. The only di↵erence between these two is the capacity size, which a↵ects the speed and security. The various padding bits and the 10⇤ 1 padding rule ensure that these six functions behave independently.
8.9
Merkle trees: using collision resistance to prove database membership
To be written.
8.10
Key derivation and the random oracle model
Although hash functions like SHA256 were initially designed to provide collision resistance, we have already seen in Section 8.7 that practitioners are often tempted to use them to solve other problems. Intuitively, hash functions like SHA256 are designed to “thoroughly scramble” their inputs, and so this approach seems to make some sense. Indeed, in Section 8.7, we looked at the problem of taking an unkeyed hash function and turning it into a keyed function that is a secure PRF, and found that it was indeed possible to give a security analysis under reasonable assumptions. In this section, we study another problem, called key derivation. Roughly speaking, the problem is this: we start with some secret data, and we want to convert it into an nbit string that we can use as the key to some cryptographic primitive, like AES. Now, the secret data may be random in some sense — at the very least, somewhat hard to guess — but it may not look anything at all like a uniformly distributed, random, nbit string. So how do we get from such a secret s to a cryptographic key t? Hashing, of course. In practice, one takes a hash function H, such as SHA256 (or, as we will ultimately recommend, some function built out of SHA256), and computes t H(s). Along the way, we will also introduce the random oracle model, which is a heuristic tool that is useful not only for analyzing the key derivation problem, but a host of other problems as well.
8.10.1
The key derivation problem
Let us look at the key derivation problem in more detail. Again, at a high level, the problem is to convert some discreet data that is hard to guess into an nbit string we can use directly as a key to some standard cryptographic primitive, such as AES. The solution in all cases will be to hash the secret to obtain the key. We begin with some motivating examples. • The secret might be a password. While such a password might be somewhat hard to guess, it could be dangerous to use such a password directly as an AES key. Even if the password were 311
uniformly distributed over a large dictionary (already a suspect assumption), the distribution of its encoding as a bit string is certainly not. It could very well that a significant fraction of passwords correspond to “weak keys” for AES that make it vulnerable to attack. Recall that AES was designed to be used with a random bit string as the key, so how it behaves on passwords is another matter entirely. • The secret could be the log of various types of system events on a running computer (e.g., the time of various interrupts such as those caused by key presses or mouse movements). Again, it might be difficult for an attacker who is outside the computer system to accurately predict the contents of such a log. However, using the log directly as an AES key is problematic: it is likely far too long, and far from uniformly distributed. • The secret could be a cryptographic key which as been partially compromised. Imagine that a user has a 128bit key, but that 64 of the bits have been leaked to the adversary. The key is still fairly difficult to guess, but it is still not uniformly distributed from the adversary’s point of view, and so should not be used directly as an AES key. • Later, we will see examples of numbertheoretic transformations that are widely used in publickey cryptography. Looking ahead a bit, we will see that for a large, composite modulus N , if x is chosen at random modulo N , and an adversary is given y := x3 mod N , it is hard to compute x. We can view x as the secret, and similarly to the previous example, we can view y as information that is leaked to the adversary. Even though the value of y completely determines x in an informationtheoretic sense, it is still widely believed to be hard to compute. Therefore, we might want to treat x as secret data in exactly the same way as in the previous examples. Many of the same issues arise here, not the least of which is that x is typically much longer (typically, thousands of bits long) than an AES key. As already mentioned, the solution that is adopted in practice is simply to hash the secret s using a hash function H to obtain the key t H(s). Let us now give a formal definition of the security property we are after. We assume the secret s is sampled according to some fixed (and publicly known) probability distribution P . We assume any such secret data can be encoded as an element of some finite set S. Further, we model the fact that some partial information about s could be leaked by introducing a function I, so that an adversary trying to guess s knows the side information I(s). Attack Game 8.2 (Guessing advantage). Let P be a probability distribution defined on a finite set S and let I be a function defined in S. For a given adversary A, the attack game runs as follows: • the challenger chooses s at random according to P and sends I(s) to A; • the adversary outputs a guess sˆ for s, and wins the game if sˆ = s. The probability that A wins this game is called its guessing advantage, and is denoted Guessadv[A, P, I]. 2 In the first example above, we might simplistically model s as being a password that is uniformly distributed over (the encodings of) some dictionary D of words. In this case, there is no
312
side information given to the adversary, and the guessing advantage is 1/D, regardless of the computational power of the adversary. In the second example above, it seems very hard to give a meaningful and reliable estimate of the guessing advantage. In the third example above, s is uniformly distributed over {0, 1}128 , and I(s) is (say) the first 64bits of s. Clearly, any adversary, no matter how powerful, has guessing advantage no greater than 2 64 . In the fourth example above, s is the number x and I(s) is the number y. Since y completely determines x, it is possible to recover s from I(s) by bruteforce search. There are smarter and faster algorithms as well, but there is no known efficient algorithm to do this. So for all efficient adversaries, the guessing advantage appears to be negligible. Now suppose we use a hash function H : S ! T to derive the key t from s. Intuitively, we want t to “look random”. To formalize this intuitive notion, we use the concept of computational indistinguishability from Section 3.11. So formally, the property that we want is that if s is sampled according to P and t is chosen at random from T , the two distributions (I(s), H(s)) and (I(s), t) are computationally indistinguishable. For an adversary A, let Distadv[A, P, I, H] be the adversary’s advantage in Attack Game 3.3 for these two distributions. The type of theorem we would like to be able to prove would say, roughly speaking, if H satisfies some specific property, and perhaps some constraints are placed on P and I, then Distadv[A, P, I, H] is not too much larger than Guessadv[A, P, I]. In fact, in certain situations it is possible prove such a theorem. We will discuss this result later, in Section 8.10.4 — for now, we will simply say that this rigorous approach is not widely used in practice, for a number of reasons. Instead, we will examine in greater detail the heuristic approach of using an “o↵ the shelf” hash function like SHA256 to derive keys. Subkey derivation. Before moving on, we consider the following, related problem: what to do with the key t derived from s. In some applications, we might use t directly as, say, and AES key. In other applications, however, we might need several keys: for example, an encryption key and a MAC key, or two di↵erent encryption keys for bidirectional secure communications (so Alice has one key for sending encrypting messages to Bob, and Bob uses a di↵erent key for sending encrypted messages to Alice). So once we have derived a single key t that “for all intents and purposes” behaves like a random bit string, we wish to derive several subkeys. We call this the subkey derivation problem to distinguish it from the key derivation problem. For the subkey derivation problem, we assume that we start with a truly random key t — it is not, but when t is computationally indistinguishable from a truly random key, this assumption is justified. Fortunately, for subkey derivation, we already have all the tools we need at our disposal. Indeed, we can derive subkeys from t using either a PRG or a PRF. For example, in the above example, if Alice and Bob have a shared key t, derived from a secret s, they can use a PRF F as follows: • derive a MAC key kmac
R
F (t, "MACKEY"); R
• derive an AlicetoBob encryption key kAB • derive a BobtoAlice encryption key kBA
R
F (t, "ABKEY");
F (t, "BAKEY").
Assuming F is a secure PRF, then the keys kmac , kAB , and kBA behave, for all intents and purposes, as independent random keys. To implement F , we can even use a hashbased PRF, like HMAC, so 313
we can do everything we need — key derivation and subkey derivation — using a single “o↵ the shelf” hash function like SHA256. So once we have solved the key derivation problem, we can use wellestablished tools to solve the subkey derivation problem. Unfortunately, the practice of using “o↵ the shelf” hash functions for key derivation is not very well understood or analyzed. Nevertheless, there are some useful heuristic models to explore.
8.10.2
Random oracles: a useful heuristic
We now introduce a heuristic that we can use to model the use of hash functions in a variety of applications, including key derivation. As we will see later in the text, this has become a popular heuristic that is used to justify numerous cryptographic constructions. The idea is that we simply model a hash function H as if it were a truly random function O. If H maps M to T , then O is chosen uniformly at random from the set Funs[M, T ]. We can translate any attack game into its random oracle version: the challenger uses O in place of H for all its computations, and in addition, the adversary is allowed to obtain the value of O at arbitrary input points of his choosing. The function O is called a random oracle and security in this setting is said to hold in the random oracle model. The function O is too large to write down and cannot be used in a real construction. Instead, we only use O as a means for carrying out a heuristic security analysis of the proposed system that actually uses H. This approach to analyzing constructions using hash function is analogous to the ideal cipher model introduced in Section 4.7, where we replace a block cipher E = (E, D) defined over (K, X ) by a family of random permutations {⇧k }k 2K . As we said, the random oracle model is used quite a bit in modern cryptography, and it would be nice to be able to use an “o↵ the shelf” hash function H, and model it as a random oracle. However, if we want a truly general purpose tool, we have to be a bit careful, especially if we want to model H as a random oracle taking variable length inputs. The basic rule of thumb is that MerkleDamg˚ ard hashes should not be used directly as general purpose random oracles. We will discuss in Section 8.10.3 how to safely (but again, heuristically) use MerkleDamg˚ ard hashes as general purpose random oracles, and we will also see that the sponge construction (see Section 8.8) can be used directly “as is”. We stress that even though security results in the random oracle are rigorous, mathematical theorems, they are still only heuristic results that do not guarantee any security for systems built with any specific hash function. They do, however, rule out “generic attacks” on systems that would work if the hash function were a random oracle. So, while such results do not rule out all attacks, they do rule out generic attacks, which is better than saying nothing at all about the security of the system. Indeed, in the real world, given a choice between two systems, S1 and S2 , where S1 comes with a security proof in the random oracle model, and S2 comes with a real security proof but is twice as slow as S1 , most practitioners would (quite reasonably) choose S1 over S2 . Defining security in the random oracle model. Suppose we have some type of cryptographic scheme S whose implementation makes use of a subroutine for computing a hash function H defined over (M, T ). The scheme S evaluates H at arbitrary points of its choice, but does not look at the internal implementation of H. We say that S uses H as an oracle. For example, Fpre (k, x) := H(k k x), which we briefly considered in Section 8.7, is a PRF that uses the hash function H as an oracle. 314
We wish to analyze the security of S. Let us assume that whatever security property we are interested in, say “property X,” is modeled (as usual) as a game between a challenger (specific to property X) and an arbitrary adversary A. Presumably, in responding to certain queries, the challenger computes various functions associated with the scheme S, and these functions may in turn require the evaluation of H at certain points. This game defines an advantage Xadv[A, S], and security with respect to property X means that this advantage should be negligible for all efficient adversaries A. If we wish to analyze S in the random oracle model, then the attack game defining security is modified so that H is e↵ectively replaced by a random function O 2 Funs[M, T ], to which both the adversary and the challenger have oracle access. More precisely, the game is modified as follows. • At the beginning of the game, the challenger chooses O 2 Funs[M, T ] at random. • In addition to its standard queries, the adversary A may submit random oracle queries: it gives m 2 M to the challenger, who responds with t = O(m). The adversary may make any number of random oracle queries, arbitrarily interleaved with standard queries. • In processing standard queries, the challenger performs its computations using O in place of H. The adversary’s advantage is defined using the same rule as before, but is denoted Xro adv[A, S] to emphasize that this is an advantage in the random oracle model. Security in the random oracle model means that Xro adv[A, S] should be negligible for all efficient adversaries A. A simple example: PRFs in the random oracle model. We illustrate how to apply the random oracle framework to construct secure PRFs. In particular, we will show that Fpre is a secure PRF in the random oracle model. We first adapt the standard PRF security game to obtain a PRF security game in the random oracle model. To make things a bit clearer, if we have a PRF F that uses a hash function H as an oracle, we denote by F O the function that uses the random oracle O in place of H. Attack Game 8.3 (PRF in the random oracle model). Let F be a PRF defined over (K, X , Y) that uses a hash function H defined over (M, T ) as an oracle. For a given adversary A, we define two experiments, Experiment 0 and Experiment 1. For b = 0, 1, we define: Experiment b: • O
R
Funs[M, T ].
• The challenger selects f 2 Funs[X , Y] as follows: if b = 0: k if b = 1: f
R R
K, f F O (k, ·); Funs[X , Y].
• The adversary submits a sequence of queries to the challenger. – F query: respond to a query x 2 X with y = f (x) 2 Y.
– Oquery: respond to a query m 2 M with t = O(m) 2 T . • The adversary computes and outputs a bit ˆb 2 {0, 1}. 315
For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to F as PRFro adv[A, F ] := Pr[W0 ] Pr[W1 ] . 2 Definition 8.3. We say that a PRF F is secure in the random oracle model if for all efficient adversaries A, the value PRFro adv[A, F ] is negligible. Consider again the PRF Fpre (k, x) := H(k k x). Let us assume that Fpre is defined over (K, X , T ), where K = {0, 1} and X = {0, 1}L , and that H is defined over (M, T ), where M includes all bit strings of length at most + L. We will show that this is a secure PRF in the random oracle model. But wait! We already argued ard hash. This seems to be in Section 8.7 that Fpre is completely insecure when H is a MerkleDamg˚ a contradiction. The problem is that, as already mentioned, it is not safe to use a MerkleDamg˚ ard hash directly as a random oracle. We will see how to fix this problem in Section 8.10.3. Theorem 8.7. If K is large then Fpre is a secure PRF when H is modeled as a random oracle. In particular, if A is a random oracle PRF adversary, as in Attack Game 8.3, that makes at most Qro oracle queries, then PRFro adv[A, Fpre ] Qro /K
Note that Theorem 8.7 is unconditional, in the sense that the only constraint on A is on the number of oracle queries: it does not depend on any complexity assumptions. Proof idea. Once H is replaced with O, the adversary has to distinguish O(k k ·) from a random function in Funs[X , T ], without the key k. Since O(k k ·) is a random function in Funs[X , T ], the only hope the adversary has is to somehow use the information returned from queries to O. We say that an Oquery k 0 k x0 is relevant if k 0 = k. It should be clear that queries to O that are not relevant cannot help distinguish O(k k ·) from random since the returned values are independent of the function O(k k ·). Moreover, the probability that after Qro queries the adversary succeeds in issuing a relevant query is at most Qro /K. 2 Proof. To make this proof idea rigorous we let A interact with two PRF challengers. For j = 0, 1, let Wj to be the event that A outputs 1 in Game j. Game 0. We write the challenger in Game 0 so that it is equivalent to Experiment 0 of Attack Game 8.3, but will be more convenient for us to analyze. We assume the adversary never makes the same Fpre query twice. Also, we use an associative array Map : M ! T to build up the random oracle on the fly, using the “faithful gnome” idea we have used so often. Here is our challenger:
316
Initialization: initialize the empty associative array Map : M ! T k R K Upon receiving an Fpre query on x 2 {0, 1}L do: t R T (1) if (k k x) 2 Domain(Map) then t Map[k k x] (2) Map[k k x] t send t to A Upon receiving an Oquery m 2 M do: t R T if m 2 Domain(Map) then t Map[m] Map[m] t send t to A It should be clear that this challenger is equivalent to that in Experiment 0 of Attack Game 8.3. In Game 0, whenever the challenger needs to sample the random oracle at some input (in processing either an Fpre query or an Oquery), it generates a random “default output”, overriding that default if it turns out the oracle has already been sampled at that input; in either case, the associative array records the input/output pair. Game 1. We make our gnome “forgetful”: we modify Game 0 by deleting the lines marked (1) and (2) in that game. Observe now that in Game 1, the challenger does not use Map or k in responding to Fpre queries: it just returns a random value. So it is clear (by the assumption that A never makes the same Fpre query twice) that Game 1 is equivalent to Experiment 1 of Attack Game 8.3, and hence PRFro adv[A, Fpre ] = Pr[W1 ] Pr[W0 ]. Let Z be the event that in Game 1, the adversary makes an Oquery at a point m = (k k x ˆ). It is clear that both games result in the same outcome unless Z occurs, so by the by Di↵erence Lemma, we have Pr[W1 ] Pr[W0 ] Pr[Z]. Since the key k is completely independent of A’s view in Game 1, each Oquery hits the key with probability 1/K, and so a simple application of the union bound yields Pr[Z] Qro /K. That completes the proof. 2 Key derivation in the random oracle model. Let us now return to the key derivation problem introduced in Section 8.10.1. Again, we have a secret s sampled from some distribution P , and information I(s) is leaked to the adversary. We want to argue that if H is modeled as a random oracle, then the adversary’s advantage in distinguishing (I(s), H(s)) from (I(s), t), where t is truly random, is not too much more than the adversary’s advantage in guessing the secret s with only I(s) (and not H(s)). To model H as a random oracle O, we convert the computational indistinguishability Attack Game 3.3 to the random oracle model, so that the attacker is now trying to distinguish
317
(I(s), O(s)) from (I(s), t), given oracle access to O. The corresponding advantage is denoted Distro adv[A, P, I, H]. Before stating our security theorem, it is convenient to generalize Attack Game 8.2 to allow the adversary to output a list of guesses sˆ1 , . . . , sˆQ , where and the adversary is said to win the game if sˆi = s for some i = 1, . . . , Q. An adversary A’s probability of winning in this game is called his list guessing advantage, denoted ListGuessadv[A, P, I]. Clearly, if an adversary A can win the above list guessing game with probability ✏, we can convert him into an adversary that wins the singleton guessing game with probability ✏/Q: we simply run A to obtain a list sˆ1 , . . . , sˆQ , choose i = 1, . . . , Q at random, and output sˆi . However, sometimes we can do better than this: using the partial information I(s) may allow us to rule out some of the sˆi ’s, and in some situations, we may be able to identify the correct sˆi uniquely. This depends on the application. Theorem 8.8. If H is modeled as a random oracle, then for every distinguishing adversary A that makes at most Qro random oracle queries, there exists a list guessing adversary B, which is an elementary wrapper around A, such that Distro adv[A, P, I, H] ListGuessadv[B, P, I] and B outputs a list of size at most Qro . In particular, there exists a guessing adversary B 0 , which is an elementary wrapper around A, such that Distro adv[A, P, I, H] Qro · Guessadv[B 0 , P, I]. Proof. The proof is almost identical to that of Theorem 8.7. We define two games, and for j = 0, 1, let Wj to be the event that A outputs 1 in Game j. Game 0. We write the challenger in Game 0 so that it is equivalent to Experiment 0 of the (I(s), H(s)) vs (H(s), t) distinguishing game. We build up the random oracle on the fly with an associative array Map : S ! T . Here is our challenger: Initialization: initialize the empty associative array Map : S ! T generate s according to P t R T (⇤) Map[s] t send (I(s), t) to A Upon receiving an Oquery sˆ 2 S do: tˆ R T if sˆ 2 Domain(Map) then tˆ Map[ˆ s] Map[ˆ s] tˆ send tˆ to A Game 1. We delete the line marked (⇤). This game is equivalent to Experiment 1 of this distinguishing game, as the value t is now truly independent of the random oracle. Moreover, both games result in the same outcome unless the adversary A in Game 1 makes an Oquery at the point s. So our list guessing adversary B simply takes the value I(s) that it receives from its own challenger, and plays the role of challenger to A as in Game 1. At the end of the game, B simply outputs Domain(Map) — the list of points at which A made Oqueries. The essential points are: 318
our B can play this role with no knowledge of s besides I(s), and it records all of the Oqueries made by A. So by the Di↵erence Lemma, we have Distro adv[A] = Pr[W0 ]
8.10.3
Pr[W1 ] ListGuessadv[B].
2
Random oracles: safe modes of operation
We have already seen that Fpre (k, x) := H(k k x) is secure in the random oracle model, and yet we know that it is completely insecure if H is a MerkleDamg˚ ard hash. The problem is that a MerkleDamg˚ ard construction has a very simple, iterative structure which exposes it to “extension attacks”. While this structure is not a problem from the point of view of collision resistance, it shows that grabbing a hash function “o↵ the shelf” and using it as if it were a random oracle is a dangerous move. In this section, we discuss how to safely use a MerkleDamg˚ ard hash as a random oracle. We will also see that the sponge construction (see Section 8.8) is already safe to use “as is”; in fact, the sponge was designed exactly for this purpose: to provide a variablelength input and variablelength output hash function that could be used directly as a random oracle. Suppose H is a MerkleDamg˚ ard hash built from a compression function h : {0, 1}n ⇥ {0, 1}` ! n {0, 1} . One recommended mode of operation is to safe HMAC with a zero key: HMAC0 (m) := HMAC(0` , m) = H(opad k H(ipad k m)). While this construction foils the obvious extension attacks, why should we have any confidence at all that HMAC0 is safe to use as a general purpose random oracle? We can only give heuristic evidence. Essentially, what we want to argue is that there are no inherent structural weaknesses in HMAC0 that give rise to a generic attack that treats the underlying compression function itself as a random oracle — or perhaps, more realistically, as a DaviesMeyer construction based on an ideal cipher. So basically, we want to show that using certain modes of operation, we can build a “big” random oracle out of a “small” random oracle — or out of an ideal cipher or even ideal permutation. This is undoubtedly a rather quixotic task — using heuristics to justify heuristics — but we shall sketch the basic ideas. The mathematical tool used to carry out such a task is called indi↵erentiability. We shall present a somewhat simplified version of this notion here. Suppose we are trying to build a “big” random oracle O out of a smaller primitive ⇢, where ⇢ could be a random oracle on a small domain, or an ideal cipher, or an ideal permutation. Let us denote by F [⇢] a particular construction for a random oracle based on the ideal primitive ⇢. Now consider a generic attack game defined by some challenger C and adversary A. Let us write the interaction between C and A as hC, Ai. We assume that the interaction results in an output bit. All of our security definitions are modeled in terms of games of this form. In the random oracle version of the attack game, with the big random oracle O, we would give both the challenger and adversary oracle access to the random function O, and we denote the interaction hCO , AO i. However, if we are using the construction F [⇢] to implement the big random oracle, then while the challenger accesses ⇢ only via the construction F , the adversary is allowed to directly query ⇢. We denote this interaction as hCF [⇢] , A⇢ i. For example, in the HMAC0 construction, the compression function h is modeled as a random oracle ⇢, or if h itself is built via DaviesMeyer, then the underlying block cipher is modeled as 319
an ideal cipher ⇢. In either case, F [⇢] corresponds to the HMAC0 construction itself. Note the asymmetry: in any attack game, the challenger only accesses ⇢ indirectly via F [⇢] (HMAC0 in this case), while the adversary can access ⇢ itself (the compression function h or the underlying block cipher). We say that F [⇢] is indi↵erentiable from O if the following holds: for every efficient challenger C and efficient adversary A, there exists an efficient adversary B, which is an elementary wrapper around A, such that Pr[hCF [⇢] , A⇢ i outputs 1]
Pr[hCO , B O i outputs 1]
is negligible. It should be clear from the definition that if we prove security of any cryptographic scheme in the random oracle model for the big random oracle O, the scheme remains secure if we implement O using F [⇢]: if an adversary A could break the scheme with F [⇢], then the adversary B above would break the scheme with O. Some safe modes. The HMAC0 construction can be proven to be indi↵erentiable from a random oracle on variable length inputs, if we either model the compression function h itself as a random oracle, or if h is built via DaviesMeyer and we model the underlying block cipher as an ideal cipher. One problem with using HMAC0 as a random oracle is that its output is fairly short. Fortunately, it is fairly easy to use HMAC0 to get a random oracle with longer outputs. Here is how. Suppose HMAC0 has an nbit output, and we need a random oracle with, say, N > n bits of output. Set q := dN/ne. Let e0 , e1 , . . . , eq be fixedlength encodings of the integers 0, 1, . . . , q. Our new hash HMAC0 (e0 k m). Then, for i = 1, . . . , q, function H 0 works as follows. On input m, we compute t HMAC0 (ei k t). Finally, we output the first N bits of t1 k t2 k · · · k tq . One we compute ti can show that H 0 is indi↵erentiable from a random oracle with N bit outputs. This result holds if we replace HMAC0 with any hash function that is itself indi↵erentiable from a random oracle with nbit outputs. Also note that when applied to long inputs, H 0 is quite efficient: it only needs to evaluate HMAC0 once on a long input. The sponge construction has been proven to be indi↵erentiable from a random oracle on variable length inputs, if we model the underlying permutation as an ideal permutation (assuming 2c , where c is the capacity is superpoly.) This includes the standardized implementations SHA3 (for fixed length outputs) and the SHAKE variants (for variable length outputs), discussed in Section 8.8.2. The special padding rules used in the SHA3 and SHAKE specifications ensure that all of the variants act as independent random oracles. Sometimes, we need random oracles whose output should be uniformly distributed over some specialized set. For example, we may want the output to be uniformly distributed over the set S = {0, . . . , d 1} for some positive integer d. To realize this, we can use a hash function H with an nbit output, which we can view as an nbit binary encoding of a number, and define H 0 (m) := H(m) mod d. If H is indi↵erentiable from a random oracle with nbit outputs, and 2n /d is superpoly, then the hash function H 0 is indi↵erentiable from a random oracle with outputs in S.
8.10.4
The leftover hash lemma
We now return to the key derivation problem. Under the right circumstances, we can solve the key derivation problem with no heuristics and no computational assumptions whatsoever. Moreover, 320
the solution is a surprising and elegant application of universal hash functions (see Section 7.1). The result, known as the leftover hash lemma, says that if we use an ✏UHF to hash a secret that can be guessed with probability at most , then provided ✏ and are sufficiently small, the output of the hash is statistically indistinguishable from a truly random value. Recall that a UHF has a key, which we normally think of as a secret key; however, in this result, the key may be made public — indeed, it could be viewed as a public, system parameter that is generated once and for all, and used over and over again. Our goal here is to simply state the result, and to indicate when and where it can (and cannot) be used. To state the result, we will need to use the notion of the statistical distance between two random variables, which we introduced in Section 3.11. Also, if s is a random variable taking values in a set S, we define the guessing probability of s to be maxx2S Pr[s = x]. Theorem 8.9 (Leftover Hash Lemma). Let H be a keyed hash function defined over (K, S, T ). Assume that H is a (1 + ↵)/N UHF, where N := T . Let k, s1 , . . . , sm be mutually independent random variables, where k is uniformly distributed over K, and each si has guessing probability at most . Let be the statistical di↵erence between (k, H(k, s1 ), . . . , H(k, sm )) and the uniform distribution on K ⇥ T m . Then we have
1 p m N + ↵. 2
Let us look at what the lemma says when m = 1. We have a secret s that can be guessed with probability at most , given whatever side information I(s) is known about s. To apply the lemma, the bound on the guessing probability must hold for all adversaries, even computationally unbounded ones. We then hash s using a random hash key k. It is essential that s (given I(s)) and k are independent — although we have not discussed the possibility here, there are potential use cases where the distribution of s or the function I can be somehow biased by an adversary in a way that depends on k, which is assumed public and known to the adversary. Therefore, to apply the lemma, we must ensure that s (given I(s)) and k are truly independent. If all of these conditions are met, then the lemma says that for any adversary A, even a computationally unbounded one, its advantage in distinguishing (k, I(s), H(k, s)) from (k, I(s), t), where t is a truly random element of T , is bounded by , as in the lemma. Now let us plug in some realistic numbers. If we want the output to be used as an AES key, we need N = 2128 . We know how to build (1/N )UHFs, so we can take ↵ = 0 (see Exercise 7.18 — with ↵ nonzero, but still quite small, one can get by with significantly shorter hash keys). If we want 2 64 , we will need the guessing probability to be about 2 256 . So in addition to all the conditions listed above, we really need an extremely small guessing probability for the lemma to be applicable. None of the examples discussed in Section 8.10.1 meet these requirements: the guessing probabilities are either not small enough, or do not hold unconditionally against unbounded adversaries, or can only be heuristically estimated. So the practical applicability to the Leftover Hash Lemma is limited — but when it does apply, it can be a very powerful tool. Also, we remark that by using the lemma with m > 1, under the right conditions, we can model the situation where the same hash key is used to derive many keys from many independent secrets with small guessing probability. The distinguishing probability grows linearly with the number of derivations, which is not surprising. 321
Because of these practical limitations, it is more typical to use cryptographic hash functions, modeled as random oracles, for key derivation, rather than UHFs. Indeed, if one uses a UHF and any of the assumptions discussed above turns out to be wrong, this could easily lead to a catastrophic security breach. Using cryptographic hash functions, while only heuristically secure for key derivation, are also more forgiving.
8.10.5
Case study: HKDF
HKDF is a key derivation function specified in RFC 5869, and is deployed in many standards. HKDF is specified in terms of the HMAC construction (see Section 8.7). So it uses the function HMAC(k, m), where k and m are variable length byte strings, which itself is implemented in terms of a MerkleDamg˚ ard hash H, such as SHA256. The input to HKDF consists of a secret s, an optional salt value salt (discussed below), an optional info field (also discussed below), and an output length parameter L. The parameters s, salt, and info are variable length byte strings. The execution of HKDF consists of two stages, called extract (which corresponds to what we called key derivation), and expand (which corresponds to what we called subkey derivation). In the extract stage, HKDF uses salt and s to compute t
HMAC(salt, s).
Using the intermediate key t, along with info, the expand (or subkey derivation) stage computes L bytes of output data, as follows: q dL/HashLene // HashLen is the output length (in bytes) of H initialize z0 to the empty string for i 1 to q do: zi HMAC(t, zi 1 k info k Octet(i)) // Octet(i) is a single byte whose value is i output the first L octets of z1 k . . . k zq When salt is empty, the extract stage of HKDF is the same as what we called HMAC0 in Section 8.10.3. As discussed there, HMAC0 can heuristically be viewed as a random oracle, and so we can use the analysis in Section 8.10.2 to show that this is a secure key derivation procedure in the random oracle model. This, if s is hard to guess, then t is indistinguishable from random. Users of HKDF have the option of providing nonzero salt. The salt plays a role akin to the random hash key used in the Leftover Hash Lemma (see Section 8.10.4); in particular, it need not be secret, and may be reused. However, it is important that the salt value is independent of the secret s and cannot be manipulated by an adversary. The idea is that under these circumstances, the output of the extract stage of HKDF seems more likely to be indistinguishable from random, without relying on the full power of the random oracle model. Unfortunately, the known security proofs apply to limited settings, so in the general case, this is still somewhat heuristic. The expand stage is just a simple application of HMAC as a PRF to derive subkeys, as we discussed at the end of Section 8.10.1. The info parameter may be used to “name” the derived subkeys, ensuring the independence of keys used for di↵erent purposes. Since the output length of the underlying hash is fixed, a simple iterative scheme is used to generate longer outputs. This stage can be analyzed rigorously under the assumption that the intermediate key t is indistinguishable from random, and that HMAC is a secure PRF — and we already know that HMAC is a secure PRF, under reasonable assumptions about the compression function of H. 322
8.11
Security without collision resistance
Theorem 8.1 shows how to extend the domain of a MAC using a collision resistant hash. It is natural to ask whether MAC domain extension is possible without relying on collision resistant functions. In this section we show that a weaker property called second preimage resistance is sufficient.
8.11.1
Second preimage resistance
We start by defining two classic security properties for nonkeyed hash functions. Let H be a hash function defined over (M, T ). • We say that H is oneway if given t := H(m) as input, for a random m 2 M, it is difficult to find an m0 2 M such that H(m0 ) = t. Such an m0 is called an inverse of t. In other words, H is oneway if it is easy to compute but difficult to invert. • We say that H is 2ndpreimage resistant if given a random m 2 M as input, it is difficult to find a di↵erent m0 2 M such that H(m) = H(m0 ). In other words, it is difficult to find an m0 that collides with a given m. • For completeness, recall that a hash function is collision resistant if it is difficult to find two distinct messages m, m0 2 M such that H(m) = H(m0 ). Definition 8.4. Let H be a hash function defined over (M, T ). We define the advantage OWadv[A, H] of an adversary A in defeating the onewayness of H as the probability of winning the following game: • the challenger chooses m 2 M at random and sends t := H(m) to A; • the adversary A outputs m0 2 M, and wins if H(m0 ) = t. H is oneway if OWadv[A, H] is negligible for every efficient adversary A. Similarly, we define the advantage SPRadv[A, H] of an adversary A in defeating the 2ndpreimage resistance of H as the probability of winning the following game: • the challenger chooses m 2 M at random and sends m to A; • the adversary A outputs m0 2 M, and wins if H(m0 ) = H(m) and m0 6= m. H is 2ndpreimage resistant if SPRadv[A, H] is negligible for every efficient adversary A. We mention some trivial relations between these notions when M is at least twice the size of T . Under this condition we have the following implications: H is collision resistant
)
H is 2ndpreimage resistant
)
H is oneway
as shown in Exercise 8.22. The converse is not true. A hash function can be 2ndpreimage resistant, but not collision resistant. For example, SHA1 is believed to be 2ndpreimage resistant even though SHA1 is not collision resistant. Similarly, a hash function can be oneway, but not be 2ndpreimage resistant. For example, the function h(x) := x2 mod N for a large odd composite N is believed to be oneway. In other words, it is believed that given x2 mod N it is difficult to find x (as long as the 323
factorization of N is unknown). However, this function H is trivially not 2ndpreimage resistant: given x 2 {1, . . . , N } as input, the value x is a second preimage since x2 mod N = ( x)2 mod N . Our goal for this section is to show that 2ndpreimage resistance is sufficient for extending the domain of a MAC and for providing file integrity. To give some intuition, consider the file integrity problem (which we discussed at the very beginning of this chapter). Our goal is to ensure that malware cannot modify a file without being detected. Recall that we hash all critical files on disk using a hash function H and store the resulting hashes in readonly memory. For a file F it should be difficult for the malware to find an F 0 such that H(F 0 ) = H(F ). Clearly, if H is collision resistant then finding such an F 0 is difficult. It would seem, however, that 2ndpreimage resistance of H is sufficient. To see why, consider malware trying to modify a specific file F without being detected. The malware is given F as input and must come up with a 2ndpreimage of F , namely an F 0 such that H(F 0 ) = H(F ). If H is 2ndpreimage resistant the malware cannot find such an F 0 and it would seem that 2ndpreimage resistance is sufficient for file integrity. Unfortunately, this argument doesn’t quite work. Our definition of 2ndpreimage resistance says that finding a 2ndpreimage for a random F in M is difficult. But files on disk are not random bit strings — it may be difficult to find a 2ndpreimage for a random file, but it may be quite easy to find a 2ndpreimage for a specific file on disk. The solution is to randomize the data before hashing it. To do so we first convert the hash function to a keyed hash function. We then require that the resulting keyed function satisfy a property called target collision resistance which we now define.
8.11.2
Randomized hash functions: target collision resistance
At the beginning of the chapter we mentioned two applications for collision resistance: extending the domain of a MAC and protecting file integrity. In this section we describe solutions to these problems that rely on a weaker security property than collision resistance. The resulting systems, although more likely to be secure, are not as efficient as the ones obtained from collision resistance. Target collision resistance. Let H be a keyed hash function. We define what it means for H to be target collision resistant, or TCR for short, using the following attack game, also shown in Fig. 8.12. Attack Game 8.4 (Target collision resistance). For a given keyed hash function H over (K, M, T ) and adversary A, the attack game runs as follows: • A sends a message m0 2 M to the challenger.
• The challenger picks a random k
R
K and sends k to A.
• A sends a second message m1 2 M to the challenger.
The adversary is said to win the game if m0 6= m1 and H(k, m0 ) = H(k, m1 ). We define A’s advantage with respect to H, denoted TCRadv[A, H], as the probability that A wins the game. 2 Definition 8.5. We say that a keyed hash function H over (K, M, T ) is target collision resistant if TCRadv[A, H] is negligible. Casting the definition in our formal mathematical framework is done exactly as for universal hash functions (Section 7.1.2). 324
k
R
K
Adversary A
TCR Challenger
m0 k m1
Figure 8.12: TCR Attack Game We note that one can view a collision resistant hash H over (M, T ) as a TCR function with an empty key. More precisely, let K be a set of size one containing only the empty word. We can define a keyed hash function H 0 over (K, M, T ) as H 0 (k, m) := H(m). It is not difficult to see that if H is collision resistant then H 0 is TCR. Thus, a collision resistant function can be viewed as the ultimate TCR hash — its key is the shortest possible.
8.11.3
TCR from 2ndpreimage resistance
We show how to build a keyed TCR hash function from a keyless 2ndpreimage resistant function such as SHA1. Let H, defined over (M, T ), be a 2ndpreimage resistant function. We construct a keyed TCR function Htcr defined over (M, M, T ) as follows: Htcr (k, m) = H(k
m)
(8.16)
Note that the length of the key k is equal to the length of the message being hashed. This is a problem for the applications we have in mind. As a result, we will only use this construction as a TCR hash for short messages. First we prove that the construction is secure. Theorem 8.10. Suppose H is 2ndpreimage resistant then Htcr is TCR. In particular, for every TCR adversary A attacking Htcr as in Attack Game 8.4, there exists a 2ndpreimage finder B, which is an elementary wrapper around A, such that TCRadv[A, Htcr ] SPRadv[B, H].
Proof. The proof is a simple direct reduction. Adversary B emulates the challenger in Attack Game 8.4 and works as follows: Input: Random m 2 M Output: m0 2 M such that m 6= m0 and H(m) = H(m0 )
1. 2. 3. 4. 5.
Run A and obtain an m0 2 M from A k m m0 Send k as the hash key to A A responds with an m1 2 M Output m0 := m1 k
We show that SPRadv[B, H] = TCRadv[A, Htcr ]. First, denote by W the event that in step (4) the messages m0 , m1 output by A are distinct and Htcr (k, m0 ) = Htcr (k, m1 ). 325
The input m given to B is uniformly distributed in M. Therefore, the key k given to A in step (2) is uniformly distributed in M and independent of A’s current view, as required in Attack Game 8.4. It follows that B perfectly emulates the challenger in Attack Game 8.4 and consequently Pr[W ] = TCRadv[A, Htcr ]. By definition of Htcr , we also have the following: Htcr (k, m0 ) = H((m Htcr (k, m1 ) = H(m1
m0 )
m0 ) = H(m)
(8.17)
0
k) = H(m )
Now, suppose event W happens. Then Htcr (k, m0 ) = Htcr (k, m1 ) and therefore, by (8.17), we know that H(m) = H(m0 ). Second, we deduce that m 6= m0 which follows since m0 6= m1 and m0 = m (m1 m0 ). Hence, when event W occurs, B outputs a 2ndpreimage of m. It now follows that: SPRadv[B, H] Pr[W ] = TCRadv[A, Htcr ] as required. 2 Target collision resistance for long inputs. The function Htcr in (8.16) shows that a 2ndpreimage resistant function directly gives a TCR function. If we assume that the SHA256 compression function h is 2ndpreimage resistant (a weaker assumption than assuming that h is collision resistant) then, by Theorem 8.10 we obtain a TCR hash for inputs of length 512 + 265 = 768 bits. The length of the required key is also 768 bits. We will often need TCR functions for much longer inputs. Using the SHA256 compression function we already know how to build a TCR hash for short inputs using a short key. Thus, let us assume that we have a TCR function h defined over (K, T ⇥ M, T ) where M := {0, 1}` for some small `, say ` = 512. We build a new TCR hash for much larger inputs. Let L 2 Z>0 be a power of 2. We build a derived TCR hash H that hashes messages in {0, 1}`L using keys in (K ⇥ T 1+log2 L ). Note that the length of the keys is logarithmic in the length of the message, which is much better than (8.16). To describe the function H we need an auxiliary function ⌫ : Z>0 ! Z>0 defined as: ⌫(x) := largest n 2 Z>0 such that 2n divides x. Thus, ⌫(x) counts the number of least significant bits of x that are zero. For example, ⌫(x) = 0 if x is odd and ⌫(x) = n if x = 2n . Note that ⌫(x) 7 for more than 99% of the integers. The derived TCR hash H is similar to MerkleDamg˚ ard. It uses the same padding block PB as in MerkleDamg˚ ard and a fixed initial value IV. The derived TCR hash H is defined as follows (see Fig. 8.13):
326
m1
IV
L
m2
h
L
k1 k2 [⌫(1)]
···
h
ms k PB
L
L
k1 k2 [⌫(2)]
h
t
k1 k2 [⌫(3)]
k2 [⌫(s)]
Figure 8.13: Extending the domain of a TCR hash Input: Message M 2 {0, 1}`L and key (k1 , k2 ) 2 K ⇥ T 1+log2 L Output: t 2 T
M M k PB Break M into consecutive `bit blocks so that M = m1 k m2 k · · · k ms where m1 , . . . , ms 2 {0, 1}` t0 IV for i = 1 to s do: u k2 [⌫(i)] ti 1 2 T ti h(k1 , (u, mi ) ) 2 T
Output ts
We note that directly using MerkleDamg˚ ard to extend the domain of a TCR hash does not work. Plugging h(k1 , ·) directly into MerkleDamg˚ ard can fail to give a TCR hash. Security of the derived hash. The following theorem shows that the derived hash H is TCR assuming the underlying hash h is. We refer to [96, 76] for the proof of this theorem. Theorem 8.11. Suppose h is a TCR hash function that hashes messages in (T ⇥ {0, 1}` ). Then, for any bounded L, the derived function H is a TCR hash for messages in {0, 1}`L . In particular, suppose A is a TCR adversary attacking H (as in Attack Game 8.4). Then there exists a TCR adversary B (whose running times are about the same as that of A) such that TCRadv[A, H] L · TCRadv[B, h]. As in MerkleDamg˚ ard this construction is inherently sequential. A treebased construction similar to Exercise 8.8 gives a TCR hash using logarithmic size keys that is more suitable for a parallel machine. We refer to [7] for the details.
8.11.4
Using target collision resistance
We now know how to build a TCR function for large inputs from a small 2ndpreimage resistant function. We show how to use such TCR functions to extend the domain for a MAC and to ensure file integrity. We start with file integrity. 327
File integrity Let H be a TCR hash defined over (K, M, T ). We use H to protect integrity of files F1 , F2 , . . . 2 M using a small amount of readonly memory. The idea is to pick a random key ri in K for every file Fi and then store the pair (ri , H(ri , Fi ) ) in readonly memory. Note that we are using a little more readonly memory than in the system based on collision resistance. To verify integrity of file Fi we simply recompute H(ri , Fi ) and compare to the hash stored in readonly memory. Why is this mechanism secure? Consider malware targeting a specific file F . We store in readonly memory the key r and t := H(r, F ). To modify F without being detected the malware must come up with a new file F 0 such that t = H(r, F 0 ). In other words, the malware is given as input the file F along with a random key r 2 K and must produce a new F 0 such that H(r, F ) = H(r, F 0 ). The adversary (the malware writer in this case) chooses which file F to attack. But this is precisely the TCR Attack Game 8.4 — the adversary chooses an F , gets a random key r, and must output a new F 0 that collides with F under r. Hence, if H is TCR the malware cannot modify F without being detected. In summary, we can provide file integrity using a small amount of readonly memory and by relying only on 2ndpreimage resistance. The cost, in comparison to the system based on collision resistance, is that we need a little more readonly memory to store the key r. In particular, using the TCR construction from the previous section, the amount of additional readonly memory needed is logarithmic in the size of the files being protected. Using a recursive construction (see Exercise 8.24) we can reduce the additional readonly memory used to a small constant, but still nonzero. Extending the domain of a MAC Let H be a TCR hash defined over (KH , M, T ). Let I = (S, V ) be a MAC for authenticating short messages in KH ⇥ T using keys in K. We assume that M is much larger than T . We build a new MAC I 0 = (S 0 , V 0 ) for authenticating messages in M using keys in K as follows: S 0 (k, m) := r h t
R
V 0 k, m, (t, r) := KH
h
H(r, m)
H(r, m)
(8.18)
Output V (k, (r, h), t)
S k, (r, h)
Output (t, r) Note the MAC signing is randomized — we pick a random TCR key r, include r in the input to the signing algorithm S, and output r as part of the final tag. As a result, tags produced by this MAC are longer than tags produced from extending MACs using a collision resistance hash (as in Section 8.2). Using the construction from the previous section, the length of r is logarithmic in the size of the message being authenticated. This extra logarithmic size key is included in every tag. On the plus side, this construction only relies on H being TCR which is a much weaker property than collision resistance and hence much more likely to hold for H. The following theorem proves security of the construction in (8.18) above. The theorem is the analog of Theorem 8.1 and its proof is similar. Note however, that the error bounds are not as tight as the bounds in Theorem 8.1. Theorem 8.12. Suppose the MAC system I is a secure MAC and the hash function H is TCR. Then the derived MAC system I 0 = (S 0 , V 0 ) defined in (8.18) is a secure MAC. 328
In particular, for every MAC adversary A attacking I 0 (as in Attack Game 6.1) that issues at most Q signing queries, there exist an efficient MAC adversary BI and an efficient TCR adversary BH , which are elementary wrappers around A, such that MACadv[A, I 0 ] MACadv[BI , I] + Q · TCRadv[BH , H].
Proof idea. Our goal is to show that no efficient MAC adversary can successfully attack I 0 . Such an adversary A asks the challenger to sign a few long messages m1 , m2 , . . . 2 M and gets back tags (ti , ri ) for i = 1, 2, . . . . It then tries to invent a new valid messageMAC pair (m, (t, r)). If A is able to produce a valid forgery (m, (t, r)) then one of two things must happen: 1. either (r, H(r, m)) is equal to (ri , H(ri , mi )) for some i; 2. or not. It is not difficult to see that forgeries of the second type can be used to attack the underlying MAC I. We show that forgeries of the first type can be used to break the target collision resistance of H. Indeed, if (r, H(r, m)) = (ri , H(ri , mi )) then r = ri and therefore H(r, m) = H(r, mi ). Thus mi and m collide under the random key r. We will show that this lets us build an adversary BH that wins the TCR game when attacking H. Unfortunately, BH must guess ahead of time which of A’s queries to use as mi . Since there are Q queries to choose from, BH will guess correctly with probability 1/Q. This is the reason for the extra factor of Q in the error term. 2 Proof. Let X be the event that adversary A wins the MAC Attack Game 6.1 with respect to I 0 . Let m1 , m2 , . . . 2 M be A’s queries during the game and let (t1 , r1 ), (t2 , r2 ), . . . be the challenger’s responses. Furthermore, let (m, (t, r)) be the adversary’s final output. We define two additional events: • Let Y denote the event that for some i = 1, 2, . . . we have that (r, H(r, m)) = (ri , H(r, mi )) and m 6= mi . • Let Z denote the event that A wins Attack Game 6.1 on I 0 and event Y did not occur. Then MACadv[A, I 0 ] = Pr[X] Pr[X ^ ¬Y ] + Pr[Y ] = Pr[Z] + Pr[Y ]
(8.19)
To prove the theorem we construct a TCR adversary BH and a MAC adversary BI such that Pr[Y ] Q · TCRadv[BH , H]
and
Pr[Z] = MACadv[BI , I].
Adversary BI is essentially the same as in the proof of Theorem 8.1. Here we only describe the TCR adversary BH , which emulates a MAC challenger for A as follows:
329
k R K u R {1, 2, . . . , Q} Run algorithm A
Upon receiving the ith signing query mi 2 M from A do: If i 6= u then ri R KH Else // i = u: for query number u get ri from the TCR challenger ˆ 0 := mi to its TCR challenger BH sends m Bh receives a random key rˆ 2 K from its challenger rˆ ri h H(ri , mi ) t S(k, (ri , h) ) Send (t, r) to A
Upon receiving the final messagetag pair (m, (t, r) ) from A do: BH sends m ˆ 1 := m to its challenger Algorithm BH responds to A’s signature queries exactly as in a real MAC attack game. Therefore, event Y happens during the interaction with BH with the same probability that it happens in a real MAC attack game. Now, when event Y happens there exists a j 2 {1, 2, . . .} such that (r, H(r, m)) = (rj , H(rj , mj )) and m 6= mj . Suppose that furthermore j = u. Then r = rj = rˆ and therefore H(ˆ r, m) = H(ˆ r, mu ). Hence, if event Y happens and j = u then BH wins the TCR attack game. In symbols, TCRadv[BH , H] = Pr[Y ^ (j = u)]. Notice that u is independent of A’s view — it is only used for choosing which random key ri is from BH ’s challenger, but no matter what u is, the key ri given to A is always uniformly random. Hence, event Y is independent of the event j = u. For the same reason, if the adversary makes a total of w queries then Pr[j = u] = 1/w 1/Q. In summary, TCRadv[BH , H] = Pr[Y ^ (j = u)] = Pr[Y ] · Pr[j = u]
Pr[Y ]/Q
as required. 2
8.12
A fun application: an efficient commitment scheme
To be written.
8.13
Another fun application: proofs of work
To be written.
8.14
Notes
Citations to the literature to be added.
330
8.15
Exercises
8.1 (Truncating a CRHF is dangerous). Let H be a collision resistant hash function defined over (M, {0, 1}n ). Use H to construct a hash function H 0 over (M, {0, 1}n ) that is also collision resistant, but if one truncates the output of H 0 by one bit then H 0 is no longer collision resistant. That is, H 0 is collision resistant, but H 00 (x) := H 0 (x)[0 . . n 2] is not. 8.2 (CRHF combiners). We want to build a CRHF H using two CRHFs H1 and H2 , so that if at some future time one of H1 or H2 is broken (but not both) then H is still secure. (a) Suppose H1 and H2 are defined over (M, T ). Let H(m) := H1 (m), H2 (m) . Show that H is a secure CRHF if either H1 or H2 is secure. (b) Show that H 0 (x) = H1 (H2 (x)) need not be a secure CRHF even if one of H1 or H2 is secure. 8.3 (Extending the domain of a PRF with a CRHF). Suppose F is a secure PRF defined over (K, X , Y) and H is a collision resistant hash defined over (M, X ). Show that F 0 (k, m) = F (k, H(m)) is a secure PRF. This shows that H can be used to extend the domain of a PRF. 8.4 (Hashthenencrypt MAC). Let H be a collision resistant hash defined over (M, X ) and let E = (E, D) be a secure block cipher defined over (K, X ). Show that the encryptedhash MAC system (S, V ) defined by S(k, m) := E(k, H(m)) is a secure MAC. Hint: Use Theorem 8.1. 8.5 (Finding many collisions). LetpH be a hash function defined over (M, T ) where N := T  and M N . We showed that O( ⇣N ) evaluations of H are sufficient to find a collision for p ⌘ H with probability 1/2. Show that O sN evaluations of H are sufficient to find s collisions (1)
(1)
(s)
(s)
(x0 , x1 ), . . . , (x0 , x1 ) for H with probability at least 1/2. Therefore, finding a million collisions is only about a thousand times harder than finding a single collision.
8.6 (Finding multicollisions). Continuing with Exercise 8.5, we say that an scollision for H is a set of s distinct points x1 , . . . , xs in M such that H(x1 ) = · · · = H(xs ). Show that for each constant value of s, O N (s 1)/s evaluations of H are sufficient to find an scollision for H, with probability at least 1/2. 8.7 (Collision finding in constant space). Let H be a hash function defined over (M, T ) where N := M. In with constant p Section 8.3 we developed a method to find an H collision p probability using O( N ) evaluations of H. However, the method required O( N ) memory space. In this exercise we develop a constantmemory collision finding method that runs in about the same time. More precisely, the method only needs memory to store two hash values in T . You may assume that H : M ! T is a random function chosen uniformly from Funs[M, T ] and T ✓ M. A collision should be produced with probability at least 1/2. (a) Let x0 R M and define H (i) (x0 ) to be the ith iterate of H starting at x0 . For example, H (3) (x0 ) = H(H(H(x0 ))). (i) Let i be the smallest positive integer satisfying H (i) (x0 ) = H (2i) (x0 ). (ii) Let j be the smallest positive integer satisfying H (j) (x0 ) = H (j+i) (x0 ). Notice that j i.
Show that H (j
1) (x
0)
and H (j+i
1) (x
0)
are an H collision with probability at least 3/4. 331
h msglen
h h
h
h m1
h m2
m3
h m4
···
···
ms
1
ms
Figure 8.14: Treebased MerkleDamg˚ ard p (b) Show that i from part p (a) satisfies i = O( N ) with probability at least 3/4 and that itpcan be found using O( N ) evaluations of H. Once i is found, finding j takes another O( N ) evaluations, as required. The entire process only needs to store two elements in T at any given time. 8.8 (A parallel MerkleDamg˚ ard). The MerkleDamg˚ ard construction in Section 8.4 gives a sequential method for extending the domain of a secure CRHF. The tree construction in Fig. 8.14 is a parallelizable approach. Prove that the resulting hash function is collision resistant, assuming h is collision resistant. Here h is a compression function h : X 2 ! X , and we assume the message length can be encoded as an element of X . 8.9 (Secure variants of DaviesMeyer). Prove that the h1 , h2 , and h3 variants of DaviesMeyer defined on page 292 are collision resistant in the ideal cipher model. 8.10 (Insecure variants of DaviesMeyer). Show that the h4 and h5 variants of DaviesMeyer defined on page 293 are not collision resistant. 8.11 (An insecure instantiation of DaviesMeyer). Let’s show that DaviesMeyer may not be collision resistant when instantiated with a realworld block cipher. Let (E, D) be a block cipher defined over (K, X ) where K = X = {0, 1}n . For y 2 X let y denote the bitwise complement of y. (a) Suppose that E(k, x) = E(k, x) for all keys k 2 K and all x 2 X . The DES block cipher has precisely this property. Show that the DaviesMeyer construction, h(k, x) := E(k, x) x, is not collision resistant when instantiated with algorithm E. (b) Suppose (E, D) is an EvenMansour cipher, E(k, x) := ⇡(x k) k, where ⇡ : X ! X is a fixed public permutation. Show that the DaviesMeyer construction instantiated with algorithm E is not collision resistant. Hint: Show that this EvenMansour cipher satisfies the property from part (a). 8.12 (MerkleDamg˚ ard without length encoding). Suppose that in the MerkleDamg˚ ard construction, we drop the requirement that the padding block encodes the message length. Let h be the compression function, let H be the resulting hash function, and let IV be the prescribed initial value.
332
(a) Show that H is collision resistant, assuming h is collision resistant and that it is hard to find a preimage of IV under h. (b) Show that if h is a DaviesMeyer compression function, and we model the underlying block cipher as an ideal cipher, then for any fixed IV, it is hard to find a preimage of IV under h. 8.13 (2ndpreimage resistance of MerkleDamg˚ ard). Let H be a MerkleDamg˚ ard hash built out of a DaviesMeyer compression function h : {0, 1}n ⇥ {0, 1}` ! {0, 1}n . Consider the attack game characterizing 2ndpreimage resistance in Definition 8.4. Let us assume that the initial, random message in that attack game consists of s blocks. We shall model the underlying block cipher used in the DaviesMeyer construction as an ideal cipher, and adapt the attack game to work in the ideal cipher model. Show that for every adversary A that makes at most Q idealcipher queries, we have (Q + s)s SPRic adv[A, H] . 2n 1 Discussion: This bound for finding second preimages is significantly better than the bound for finding arbitrary collisions. Unfortunately, we have to resort to the ideal cipher model to prove it. 8.14 (Fixed points). We consider the DaviesMeyer and MiyaguchiPreneel compression functions defined in Section 8.5.2. (a) Show that for a DaviesMeyer compression function it is easy to find a pair (t, m) such that hDM (t, m) = t. Such a pair is called a fixed point for hDM . (b) Show that in the ideal cipher model it is difficult to find fixed points for the MiyaguchiPreneel compression function. The next exercise gives an application for fixed points. 8.15 (Finding second preimages in MerkleDamg˚ ard). In this exercise, we develop a second preimage attack on MerkleDamg˚ ard that roughly matches the security bounds in Exercise 8.13. ard hash built out of a DaviesMeyer compression function h : {0, 1}n ⇥ Let HMD be a MerkleDamg˚ {0, 1}` ! {0, 1}n . Recall that HMD pads a given message with a padding block that encodes the message length. We will also consider the hash function H, which is the same as HMD , but which uses a padding block that does not encode the message length. Throughout this exercise, we model the underlying block cipher in the DaviesMeyer construction as an ideal cipher. For concreteness, assume ` = 2n. (a) Let s ⇡ 2n/2 . You are given a message M that consists of s random `bit blocks. Show that by making O(s) ideal cipher queries, with probability 1/2 you can find a message M 0 6= M such that H(M 0 ) = H(M ). Here, the probability is over the random choice of M , the random permutations defining the ideal cipher, and the random choices made by your attack. Hint: Repeatedly choose random blocks x in {0, 1}` until h(IV, x) is the same as one of the s chaining variables obtained when computing H(M ). Use this x to construct the second preimage M 0 . (b) Repeat part (a) for HMD . Hint: The attack in part (a) will likely find a second preimage M 0 that is shorter than M ; because of length encoding, this will not be a second preimage under HMD ; nevertheless, show 333
how to use fixed points (see previous exercise) to modify M 0 so that it has the same length as M . Discussion: Let H be a hash function with an nbit output. If H is a random function then breaking second preimage resistance takes about 2n time. This exercise shows that for MerkleDamg˚ ard functions, breaking second preimage resistance can be done much faster, taking only about 2n/2 time. 8.16 (The envelope method is a secure PRF). Consider the envelope method for building a PRF from a hash function discussed in Section 8.7: Fenv (k, M ) := H(k k M k k). Here, we assume that H is a MerkleDamg˚ ard hash built from a compression function h : {0, 1}n ⇥ {0, 1}` ! {0, 1}n . Assume that the keys for Fenv are `bit strings. Furthermore, assume that the message M a bit string whose length is an even multiple of ` (we can always pad the message, if necessary). Under the assumption that both htop and hbot are secure PRFs, show that Fenv is a secure PRF. Hint: Use the result of Exercise 7.6; also, first consider a simplified setting where H does not append the usual MerkleDamg˚ ard padding block to the inputs k k M k k (this padding block does not really help in this setting, but it does not hurt either — it just complicates the analysis). 8.17 (The keyprepending method revisited). Consider the keyprepending method for building a PRF from a hash function discussed in Section 8.7: Fpre (k, M ) := H(k k M ). Here, we assume that H is a MerkleDamg˚ ard hash built from a compression function h : {0, 1}n ⇥ {0, 1}` ! {0, 1}n . Assume that the keys for Fpre are `bit strings. Under the assumption that both htop and hbot are secure PRFs, show that Fpre is a prefixfree secure PRF. 8.18 (The keyappending method revisted). Consider the following variant of the key0 (k, M ) := appending method for building a PRF from a hash function discussed in Section 8.7: Fpost H(M k PB k k). Here, we assume that H is a MerkleDamg˚ ard hash built from a compression ard padding for function h : {0, 1}n ⇥ {0, 1}` ! {0, 1}n . Also, PB is the standard MerkleDamg˚ 0 are `bit strings. Under the M , which encodes the length of M . Assume that the keys for Fpost 0 is a secure PRF. assumption that h is collision resistant and htop is a secure PRF, show that Fpost 8.19 (Dual PRFs). The security analysis of HMAC assumes that the underlying compression function is a secure PRF when either input is used as the key. A PRF with this property is said to be a dual PRF. Let F be a secure PRF defined over (K, X , Y) where Y = {0, 1}n for some n. We wish to build a new PRF Fˆ that is a dual PRF. This Fˆ can be used as a building block for HMAC. (a) Suppose K = X . Show that the most natural construction Fˆ (x, y) := F (x, y) F (y, x) is insecure: there exists a secure PRF F for which Fˆ is not a dual PRF. Hint: Start from a secure PRF F 0 and the “sabotage” it to get the required F . (b) Let G be a PRG defined over (S, K ⇥ X ). Let G0 : S ! K be the left output of G and let G1 : S ! X be the right output of G. Let Fˆ be the following PRF defined over (S, S, Y): ⌘ ⇣ ⌘ ⇣ F G0 (y), G1 (x) . Fˆ (x, y) := F G0 (x), G1 (y) Prove that Fˆ is a dual PRF assuming G is a secure PRG and that G1 is collision resistant.
8.20 (Sponge with low capacity is insecure). Let H be a sponge hash with rate r and capacity c, built from a permutation ⇡ : {0, 1}n ! {0, 1}n , where n = r + c (see Section 8.8). 334
Assume r 2c. Show how to find a collision for H with probability at least 1/2 in time O(2c/2 ). The colliding messages can be 2r bits each. 8.21 (Sponge as a PRF). Let H be a sponge hash with rate r and capacity c, built from a permutation ⇡ : {0, 1}n ! {0, 1}n , where n = r + c (see Section 8.8). Consider again the PRF built from H by prepending the key: Fpre (k, M ) := H(k k M ). Assume that the key is r bits and the output of Fpre is also r bits. Prove that in the ideal permutation model, where ⇡ is replaced by a random permutation ⇧, this construction yields a secure PRF, assuming 2r and 2c are superpoly. Note: This follows immediately from the fact that H is indi↵erentiable from a random oracle (see Section 8.10.3) and Theorem 8.7. However, you are to give a direct proof of this fact. Hint: Use the same domain splitting strategy as outlined in Exercise 7.17. 8.22 (Relations among definitions). Let H be a hash function over (M, T ) where M 2T . We say that an element m 2 M has a second preimage if there exists a di↵erent m0 2 M such that H(m) = H(m0 ). (a) Show that at least half the elements of M have a second preimage. (b) Use part (a) to show that a 2ndpreimage hash must be oneway. (c) Show that a collision resistant hash must be 2ndpreimage resistant. 8.23 (From TCR to 2ndpreimage resistance). Let H be a TCR hash defined over (K, M, T ). Choose a random r 2 M. Prove that f (x) := H(r, x) k r is 2ndpreimage resistant, where r is treated as a system parameter. 8.24 (File integrity: reducing readonly memory). The file integrity construction in Section 8.11.4 uses additional readonly memory proportional to log F  where F  is the size of the file F being protected. (a) By first hashing the file F and then hashing the key r, show how to reduce the amount of additional readonly memory used to O(log log F ). This requires storing additional O(log F ) bits on disk. (b) Generalize your solution from part (a) to show how to reduce readonly overhead to constant size independent of F . The extra information stored on disk is still of size O(log F ). 8.25 (Strong 2ndpreimage resistance). Let H be a hash function defined over (X ⇥ Y, T ) where X := {0, 1}n . We say that H is strong 2ndpreimage resistant, or simply strongSPR, if no efficient adversary, given a random x in X as input, can output y, x0 , y 0 such that H(x, y) = H(x0 , y 0 ) with nonnegligible probability. (a) Let H be a strongSPR. Use H to construct a collision resistant hash function H 0 defined over (X ⇥ Y, T ). (b) Let us show that a function H can be a strongSPR, but not collision resistant. For example, consider the hash function: H 00 (0, 0) := H 00 (0, 1) := 0
and
H 00 (x, y) := H(x, y) for all other inputs.
Prove that if X  is superpoly and H is a strongSPR then so is H 00 . However, H 00 is clearly not collision resistant. 335
(c) Show that HTCR (k, (x, y)) := H((k SPR hash function.
x), y) is a TCR hash function assuming H is a strong
8.26 (Enhanced TCR). Let H be a keyed hash function defined over (K, M, T ). We say that H is an enhancedTCR if no efficient adversary can win the following game with nonnegligible advantage: the adversary outputs m 2 M, is given random k 2 K and outputs (k 0 , m0 ) such that H(k, m) = H(k 0 , m0 ). (a) Let H be a strongSPR hash function over (X ⇥ Y, T ), as defined in Exercise 8.25, where X := {0, 1}n . Show that H 0 (k, (x, y)) := H((k x), y) is an enhancedTCR hash function. (b) Show how to use an enhancedTCR to extend the domain of a MAC. Let H be a enhancedTCR defined over (KH , M, X ) and let (S, V ) be a secure MAC defined over (K, X , T ). Show that the following is a secure MAC: S(k, H(r, m)), output (r, t)} S 0 (k, m) := { r R KH , t 0 V k, m, (r, t) := { accept if t = V (k, H(r, m))} 8.27 (Weak collision resistance). Let H be a keyed hash function defined over (K, M, T ). We say that H is a weak collision resistant (WCR) if no efficient adversary can win the following game with nonnegligible advantage: the challenger chooses a random key k 2 K and lets the adversary query the function H(k, ·) at any input of its choice. The adversary wins if it outputs a collision m0 , m1 for H(k, ·). (a) Show that WCR is a weaker notion than a secure MAC: (1) show that every deterministic secure MAC is WCR, (2) give an example of a secure WCR that is not a secure MAC. (b) MAC domain extension with a WCR: let (S, V ) be a secure MAC and let H be a WCR. Show that the MAC system (S 0 , V 0 ) defined by S 0 (k0 , k1 ), m := S k1 , H(k0 , m) is secure. (c) Show that MerkleDamg˚ ard expands a compressing fixedinput length WCR to a variable input length WCR. In particular, let h be a WCR defined over (K, X ⇥ Y, X ), where X := {0, 1}n and Y := {0, 1}` . Define H as a keyed hash function over (K, {0, 1}L , X ) as follows: 9 8 pad and break M into `bit blocks: m1 , . . . , ms > > > > > > n 2X > > t 0 > > 0 > > > > > > = < for i = 1 to s do: ti h k1 , (ti 1 , mi ) H (k1 , k2 ), M := > > > > > > > > encode s as a block b 2 Y > > > > > > h k , (t , b) t s+1 2 s > > ; : output ts+1 Show that H is a WCR if h is.
8.28 (The trouble with random oracles). Let H be a hash function defined over (K ⇥ X , Y). We showed that H(k, x) is a secure PRF when H is modeled as a random oracle. In this exercise we show that this PRF can be tweaked into a new PRF F that uses H as a blackbox, and that is a secure PRF when H is modeled as a random model. However, for every concrete instantiation of the hash function H, the PRF F becomes insecure. 336
For simplicity, assume that K and Y consist of bit strings of length n and that X consists of bit strings of length at most L for some polybounded n and L. Assume also that the program for H parses its input as a bit string of the form k k x, where k 2 K and x 2 X . Consider a program Exec(P, v, t) that takes as input three bit strings P, v, t. When Exec(P, v, t) runs, it attempts to interpret P as a program written in some programming language (take your pick); it runs P on input v, but stops the execution after t steps (if necessary), where t is the bitlength of t. The output of Exec(P, v, t) is whatever P outputs on input v, or some special default value if the time bound is exceeded. For simplicity, assume that Exec(P, v, t) always outputs an nbit string (padding or truncating as necessary). Even though P on input v may run in exponential time (or even fall into an infinite loop), Exec(P, v, t) always runs in time bounded by a polynomial in its input length. Finally, let T be some arbitrary polynomial, and define F (k, x) := H(k, x)
Exec(x, k k x, 0T (k+x) ).
(a) Show that if H is any hash function that can be implemented by a program PH whose length is at most L and whose running time on input k k x is at most T (k + x), then the concrete instantiation of F using this H runs in polynomial time and is not a secure PRF. Hint: Find a value of x that makes the PRF output 0n , for all keys k 2 K. (b) Show that F is a secure PRF if H is modeled as a random oracle. Discussion: Although this is a contrived example, it shakes our confidence in the random oracle model. Nevertheless, the reason why the random oracle model has been so successful in practice is that typically realworld attacks treat the hash function as a black box. The attack on F clearly does not. See also the discussion in [24], which removes the strict time bound restriction on H.
337
Chapter 9
Authenticated Encryption Our discussion of encryption in Chapters 2 to 8 leads up to this point. In this chapter we, construct systems that ensure both data secrecy (confidentiality) and data integrity, even against very aggressive attackers that can interact with the sender and receiver quite maliciously and arbitrarily. Such systems are said to provide authenticated encryption or are simply said to be AEsecure. This chapter concludes our discussion of symmetric encryption. It is the culmination of our symmetric encryption story. Recall that in our discussion of CPA security in Chapter 5 we stressed that CPA security does not provide any integrity. An attacker can tamper with the output of a CPAsecure cipher without being detected by the decryptor. We will present many realworld settings where undetected ciphertext tampering comprises both message secrecy and message integrity. Consequently, CPA security by itself is insufficient for almost all applications. Instead, applications should almost always use authenticated encryption to ensure both message secrecy and integrity. We stress that even if secrecy is the only requirement, CPA security is insufficient. In this chapter we develop the notion of authenticated encryption and construct several AE systems. There are two general paradigms for construction AE systems. The first, called generic composition, is to combine a CPAsecure cipher with a secure MAC. There are many ways to combine these two primitives and not all combinations are secure. We briefly consider two examples. Let (E, D) be a cipher and (S, V ) be a MAC. Let kenc be a cipher key and kmac be a MAC key. Two options for combining encryption and integrity immediately come to mind, which are shown in Fig. 9.1 and work as follows: EncryptthenMAC Encrypt the message, c R E(kenc , m), then MAC the ciphertext, tag R S(kmac , c); the result is the ciphertexttag pair (c, tag). This method is supported in the TLS 1.2 protocol and later versions as well as in the IPsec protocol and in a widelyused NIST standard called GCM (see Section 9.7). MACthenencrypt MAC the message, tag R S(kmac , m), then encrypt the messagetag pair, c R E kenc , (m, t) ; the result is the ciphertext c. This method is used in older versions of TLS (e.g., SSL 3.0 and its successor called TLS 1.0) and in the 802.11i WiFi encryption protocol. As it turns out, only the first method is secure for every combination of CPAsecure cipher and secure MAC. The intuition is that the MAC on the ciphertext prevents any tampering with the ciphertext. We will show that the second method can be insecure — the MAC and cipher can 338
m
m tag
c
E(kenc , m ) tag
S(kmac , m) m
tag
S(kmac , c) c
c
tag
encryptthenmac
E(kenc , (m, tag) )
macthenencrypt
Figure 9.1: Two methods to combine encryption and MAC interact badly and cause the resulting system to not be AEsecure. This has lead to many attacks on widely deployed systems. The second paradigm for building authenticated encryption is to build them directly from a block cipher or a PRF without first constructing either a standalone cipher or MAC. These are sometimes called integrated schemes. The OCB encryption mode is the primary example in this category (see Exercise 9.17). Other examples include IAPM, XCBC, CCFB, and others. Authenticated encryption standards. Cryptographic libraries such as OpenSSL often provide an interface for CPAsecure encryption (such as counter mode with a random IV) and a separate interface for computing MACs on messages. In the past, it was up to developers to correctly combine these two primitives to provide authenticated encryption. Every system did it di↵erently and not all incarnations used in practice were secure. More recently, several standards have emerged for secure authenticated encryption. A popular method called Galois Counter Mode (GCM) uses encryptthenMAC to combine random counter mode encryption with a CarterWegman MAC (see Section 9.7). We will examine the details of this construction and its security later on in the chapter. Developers are encouraged to use an authenticated encryption mode provided by the underlying cryptographic library and to not implement it themselves.
9.1
Authenticated encryption: definitions
We start by defining what it means for a cipher E to provide authenticated encryption. It must satisfy two properties. First, E must be CPAsecure. Second, E must provide ciphertext integrity, as defined below. Ciphertext integrity is a new property that captures the fact that E should have properties similar to a MAC. Let E = (E, D) be a cipher defined over (K, M, C). We define ciphertext integrity using the following attack game, shown in Fig. 9.2. The game is analogous to the MAC Attack Game 6.1. Attack Game 9.1 (ciphertext integrity). For a given cipher E = (E, D) defined over (K, M, C), and a given adversary A, the attack game runs as follows: • The challenger chooses a random k
R
K.
339
Adversary A
Challenger k
R
K
mi ci
E(k, mi ) c
Figure 9.2: Ciphertext integrity game (Attack Game 9.1) • A queries the challenger several times. For i = 1, 2, . . . , the ith query consists of a message mi 2 M. The challenger computes ci R E(k, mi ), and gives ci to A.
• Eventually A outputs a candidate ciphertext c 2 C that is not among the ciphertexts it was given, i.e., c 62 {c1 , c2 , . . .}. We say that A wins the game if c is a valid ciphertext under k, that is, D(k, c) 6= reject. We define A’s advantage with respect to E, denoted CIadv[A, E], as the probability that A wins the game. Finally, we say that A is a Qquery adversary if A issues at most Q encryption queries. 2 Definition 9.1. We say that a E = (E, D) provides ciphertext integrity, or CI for short, if for every efficient adversary A, the value CIadv[A, E] is negligible. CPA security and ciphertext integrity are the properties needed for authenticated encryption. This is captured in the following definition. Definition 9.2. We say that a cipher E = (E, D) provides authenticated encryption, or is simply AEsecure, if E is (1) semantically secure under a chosen plaintext attack, and (2) provides ciphertext integrity. Why is Definition 9.2 the right definition? In particular, why are we requiring ciphertext integrity, rather than some notion of plaintext integrity (which might seem more natural)? In Section 9.2, we will describe a very insidious class of attacks called chosen ciphertext attacks, and we will see that our definition of AEsecurity is sufficient (and, indeed, necessary) to prevent such attacks. In Section 9.3, we give a more highlevel justification for the definition. Onetime authenticated encryption In practice, one often uses a symmetric key to encrypt a single message. The key is never used again. For example, when sending encrypted email one often picks an ephemeral key and encrypts the email body under this ephemeral key. The ephemeral key is then encrypted and transmitted in the email header. A new ephemeral key is generated for every email. In these settings one can use a onetime encryption scheme such as a stream cipher. The cipher must be semantically secure, but need not be CPAsecure. Similarly, it suffices that the cipher provide onetime ciphertext integrity, which is a weaker notion than ciphertextintegrity. In 340
particular, we change Attack Game 9.1 so that the adversary can only obtain the encryption of a single message m. Definition 9.3. We say that E = (E, D) provides onetime ciphertext integrity if for every efficient singlequery adversary A, the value CIadv[A, E] is negligible. Definition 9.4. We say that E = (E, D) provides onetime authenticated encryption, or is 1AEsecure for short, if E is semantically secure and provides onetime ciphertext integrity. In applications that only use a symmetric key once, 1AEsecurity suffices. We will show that the encryptthenMAC construction of Fig. 9.1 using a semantically secure cipher and a onetime MAC, provides onetime authenticated encryption. Replacing the MAC by a onetime MAC can lead to efficiency improvements.
9.2
Implications of authenticated encryption
Before constructing AEsecure systems, let us first play with Definition 9.1 a bit to see what it implies. Consider a sender, Alice, and a receiver, Bob, who have a shared secret key k. Alice sends a sequence of messages to Bob over a public network. Each message is encrypted with an AEsecure cipher E = (E, D) using the key k. For starters, consider an eavesdropping adversary A. Since E is CPAsecure this does not help A learn any new information about messages sent from Alice to Bob. Now consider a more aggressive adversary A that attempts to make Bob receive a message that was not sent by Alice. We claim this cannot happen. To see why, consider the following singlemessage example: Alice encrypts to Bob a message m and the resulting ciphertext c is intercepted by A. The adversary’s goal is to create some cˆ such that m ˆ := D(k, cˆ) 6= reject and m ˆ 6= m. This cˆ would fool Bob into thinking that Alice sent m ˆ rather than m. But then A could also win Attack Game 9.1 with respect to E, contradicting E’s ciphertext integrity. Consequently, A cannot modify c without being detected. More generally, applying the argument to multiple messages shows that A cannot cause Bob to receive any messages that were not sent by Alice. The more general conclusion here is that ciphertext integrity implies message integrity.
9.2.1
Chosen ciphertext attacks: a motivating example
We now consider an even more aggressive type of attack, called a chosen ciphertext attack for short. As we will see, an AEsecure cipher provides message secrecy and message integrity even against such a powerful attack. To motivate chosen ciphertext attacks suppose Alice sends an email message to Bob. For simplicity let us assume that every email starts with the letters To: followed by the recipient’s email address. So, an email to Bob starts with To:[email protected] and an email to Mel begins with To:[email protected]. The mail server decrypts every incoming email and writes it into the recipient’s inbox: emails that start with To:[email protected] are written to Bob’s inbox and emails that start with To:[email protected] are written to Mel’s inbox. Mel, the attacker in this story, wants to read the email that Alice sent to Bob. Unfortunately for Mel, Alice was careful and encrypted the email using a key known only to Alice and to the mail server. When the ciphertext c is received at the mail server it will be decrypted and the resulting message is placed into Bob’s inbox. Mel will be unable to read it. 341
Nevertheless, let us show that if Alice encrypts the email with a CPAsecure cipher such as randomized counter mode or randomized CBC mode then Mel can quite easily obtain the email contents. Here is how: Mel will intercept the ciphertext c enroute to the mail server and modify it to obtain a ciphertext cˆ so that the decryption of cˆ starts with To:[email protected], but is otherwise the same as the original message. Mel then forwards cˆ to the mail server. When the mail server receives cˆ it will decrypt it and (incorrectly) place the plaintext into Mel’s inbox where Mel can easily read it. To successfully carry out this attack, Mel must first solve the following problem: given an encryption c of some message (u k m) where u is a fixed known prefix (in our case u := To:[email protected]), compute a ciphertext cˆ that will decrypt to the message (v k m), where v is some other prefix (in our case v := To:[email protected]). Let us show that Mel can easily solve this problem, assuming the encryption scheme is either randomized counter mode or randomized CBC. For simplicity, we also assume that u and v are binary strings whose length is the same as the block size of the underlying block cipher. As usual c[0] and c[1] are the first and second blocks of c where c[0] is the random IV. Mel constructs cˆ as follows: • randomized counter mode: define cˆ to be the same as c except that cˆ[1] := c[1] • randomized CBC mode: define cˆ to be the same as c except that cˆ[0] := c[0]
u u
v. v.
It is not difficult to see that in either case the decryption of cˆ starts with the prefix v (see Section 3.3.2). Mel is now able to obtain the decryption of cˆ and read the secret message m in the clear. What just happened? We proved that both encryption modes are CPA secure, and yet we just showed how to break them. This attack is an example of a chosen ciphertext attack — by querying for the decryption of cˆ, Mel was able to deduce the decryption of c. This attack is also another demonstration of how attackers can exploit the malleability of a cipher — we saw another attack based on malleability back in Section 3.3.2. As we just saw, a CPAsecure system can become completely insecure when an attacker can decrypt certain ciphertexts, even if he cannot directly decrypt a ciphertext that interests him. Put another way, the lack of ciphertext integrity can completely compromise secrecy — even if plaintext integrity is not an explicit security requirement. We informally argue that if Alice used an AEsecure cipher E = (E, D) then it would be impossible to mount the attack we just described. Suppose Mel intercepts a ciphertext c := E(k, m). He tries to create another ciphertext cˆ such that (1) m ˆ := D(k, cˆ) starts with prefix v, and (2) the adversary can recover m from m, ˆ in particular m ˆ 6= reject. Ciphertext integrity, and therefore AEsecurity, implies that the attacker cannot create this cˆ. In fact, the attacker cannot create any new valid ciphertexts and therefore an AEsecure cipher foils the attack. In the next section, we formally define the notion of a chosen ciphertext attack, and show that if a cipher is AEsecure then it is secure even against this type of attack.
9.2.2
Chosen ciphertext attacks: definition
In this section, we formally define the notion of a chosen ciphertext attack. In such an attack, the adversary has all the power of an attacker in a chosen plaintext attack, but in addition, the 342
adversary may obtain decryptions of ciphertexts of its choosing — subject to a restriction. Recall that in a chosen plaintext attack, the adversary obtains a number of ciphertexts from its challenger, in response to encryption queries. The restriction we impose is that the adversary may not ask for the decryptions of any of these ciphertexts. While such a restriction is necessary to make the attack game at all meaningful, it may also seem a bit unintuitive: if the adversary can decrypt ciphertexts of choosing, why would it not decrypt the most important ones? We will explain later (in Section 9.3) more of the intuition behind this definition. We will show below (in Section 9.2.3) that if a cipher is AEsecure then it is secure against chosen ciphertext attack. Here is the formal attack game: Attack Game 9.2 (CCA security). For a given cipher E = (E, D) defined over (K, M, C), and for a given adversary A, we define two experiments. For b = 0, 1, we define Experiment b: • The challenger selects k
R
K.
• A then makes a series of queries to the challenger. Each query can be one of two types: – Encryption query: for i = 1, 2, . . . , the ith encryption query consists of a pair of messages (mi0 , mi1 ) 2 M2 . The challenger computes ci R E(k, mib ) and sends ci to A.
– Decryption query: for j = 1, 2, . . . , the jth decryption query consists of a ciphertext cˆj 2 C that is not among the responses to the previous encryption queries, i.e., cˆj 2 / {c1 , c2 , . . .}. The challenger computes m ˆj
D(k, cˆj ), and sends m ˆ j to A.
• At the end of the game, the adversary outputs a bit ˆb 2 {0, 1}. Let Wb is the event that A outputs 1 in Experiment b and define A’s advantage with respect to E as CCAadv[A, E] := Pr[W0 ] Pr[W1 ] . 2 We stress that in the above attack game, the encryption and decryption queries may be arbitrarily interleaved with one another. Definition 9.5 (CCA security). A cipher E is called semantically secure against chosen ciphertext attack, or simply CCAsecure, if for all efficient adversaries A, the value CCAadv[A, E] is negligible. In some settings, a new key is generated for every message so that a particular key k is only used to encrypt a single message. The system needs to be secure against chosen ciphertext attacks where the attacker fools the user into decrypting multiple ciphertexts using k. For these settings we define security against an adversary that can only issue a single encryption query, but many decryption queries. Definition 9.6 (1CCA security). In Attack Game 9.2, if the adversary A is restricted to making a single encryption query, we denote its advantage by 1CCAadv[A, E]. A cipher E is onetime semantically secure against chosen ciphertext attack, or simply, 1CCAsecure, if for all efficient adversaries A, the value 1CCAadv[A, E] is negligible. 343
As discussed in Section 2.3.5, Attack Game 9.2 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A. In this game, we measure A’s bitguessing advantage CCAadv⇤ [A, E] (and 1CCAadv⇤ [A, E]) as Pr[ˆb = b] 1/2. The general result of Section 2.3.5 (namely, (2.13)) applies here as well: CCAadv[A, E] = 2 · CCAadv⇤ [A, E].
(9.1)
And similarly, for adversaries restricted to a single encryption query, we have: 1CCAadv[A, E] = 2 · 1CCAadv⇤ [A, E].
9.2.3
(9.2)
Authenticated encryption implies chosen ciphertext security
We now show that every AEsecure system is also CCAsecure. Similarly, every 1AEsecure system is 1CCAsecure. Theorem 9.1. Let E = (E, D) be a cipher. If E is AEsecure, then it is CCAsecure. If E is 1AEsecure, then it is 1CCAsecure. In particular, suppose A is a CCAadversary for E that makes at most Qe encryption queries and Qd decryption queries. Then there exist a CPAadversary Bcpa and a CIadversary Bci , where Bcpa and Bci are elementary wrappers around A, such that CCAadv[A, E] CPAadv[Bcpa , E] + 2Qd · CIadv[Bci , E].
(9.3)
Moreover, Bcpa and Bci both make at most Qe encryption queries.
Before proving this theorem, we point out a converse of sorts: if a cipher is CCAsecure and provides plaintext integrity, then it must be AEsecure. You are asked to prove this in Exercise 9.15. These two results together provide strong support for the claim that AEsecurity is the right notion of security for general purpose communication over an insecure network. We also note that it is possible to build a CCAsecure cipher that does not provide ciphertext (or plaintext) integrity — see Exercise 9.12 for an example. Proof idea. A CCAadversary A issues encryption and allowed decryption queries. We first argue that the response to all these decryption queries must be reject. To see why, observe that if the adversary ever issues a valid decryption query ci whose decryption is not reject, then this ci can be used to win the ciphertext integrity game. Hence, since all of A’s decryption queries are rejected, the adversary learns nothing by issuing decryption queries and they may as well be discarded. After removing decryption queries we end up with a standard CPA game. The adversary cannot win this game because E is CPAsecure. We conclude that A has negligible advantage in winning the CCA game. 2 Proof. Let A be an efficient CCAadversary attacking E as in Attack Game 9.2, and which makes at most Qe encryption queries and Qd decryption queries. We want to show that CCAadv[A, E] is negligible, assuming that E is AEsecure. We will use the bitguessing versions of the CCA and CPA attack games, and show that CCAadv⇤ [A, E] CPAadv⇤ [Bcpa , E] + Qd · CIadv[Bci , E]. 344
(9.4)
for efficient adversaries Bcpa and Bci . Then (9.3) follows from (9.4), along with (9.1) and (5.4). Moreover, as we shall see, the adversary Bcpa makes at most Qe encryption queries; therefore, if E is 1AEsecure, it is also 1CCAsecure. Let us define Game 0 to be the bitguessing version of Attack Game 9.2. The challenger in this game, called Game 0, works as follows: b k
R R
{0, 1} K
//
A will try to guess b
upon receiving the ith encryption query (mi0 , mi1 ) from A do: send ci R E(k, mb ) to A (1)
upon receiving the jth decryption query cˆj from A do: send D(k, cˆj ) to A
Eventually the adversary outputs a guess ˆb 2 {0, 1}. We say that A wins the game if b = ˆb and we denote this event by W0 . By definition, the bitguessing advantage is CCAadv⇤ [A, E] = Pr[W0 ]
1/2.
(9.5)
Game 1. We now modify line (1) in the challenger as follows: (1)
send reject to A
We argue that A cannot distinguish this challenger from the original. Let Z be the event that in Game 1, A issues a decryption query cˆj such that D(k, cˆj ) 6= reject. Clearly, Games 0 and 1 proceed identically as long as Z does not happen. Hence, by the Di↵erence Lemma (i.e., Theorem 4.7) it follows that Pr[W0 ] Pr[W1 ] Pr[Z]. Using a “guessing strategy” similar to that used in the proof of Theorem 6.1, we can use A to build a CIadversary Bci that wins the CI attack game with probability at least Pr[Z]/Qd . Note that in Game 1, the decryption algorithm is not used at all. Adversary Bci ’s strategy is simply to guess a random number ! 2 {1, . . . , Qd }, and then to play the role of challenger to A: • when A makes an encryption query, Bci forwards this to its own challenger, and returns the response to A; • when A makes a decryption query cˆj , Bci simply sends reject to A, except that if j = !, Bci outputs cˆj and halts. It is not hard to see that CIadv[Bci , E] Pr[W0 ]
Pr[Z]/Qd , and so
Pr[W1 ] Pr[Z] Qd · CIadv[Bci , E].
(9.6)
Final reduction. Since all decryption queries are rejected in Game 1, this is essentially a CPA attack game. More precisely, we can construct a CPA adversary Bcpa that plays the role of challenger to A as follows: • when A makes an encryption query, Bcpa forwards this to its own challenger, and returns the response to A;
• when A makes a decryption query, Bcpa simply sends reject to A. At the end of the game, Bcpa simply outputs the bit ˆb that A outputs. Clearly, Pr[W1 ]
1/2 = CPAadv⇤ [Bcpa , E]
Putting equations (9.5)–(9.7) together gives us (9.4), which proves the theorem. 2 345
(9.7)
9.3
Encryption as an abstract interface
To further motivate the definition of authenticated encryption we show that it precisely captures an intuitive notion of secure encryption as an abstract interface. AEsecurity implies that the real implementation of this interface may be replaced by an idealized implementation in which messages literally jump from sender to receiver, without going over the network at all (even in encrypted form). We now develop this idea more fully. Suppose a sender S and receiver R are using some arbitrary Internetbased system (e.g, gambling, auctions, banking — whatever). Also, we assume that S and R have already established a shared, random encryption key k. During the protocol, S will send encryptions of messages m1 , m2 , . . . to R. The messages mi are determined by the logic of the protocol S is using, whatever that happens to be. We can imagine S placing a message mi in his “outbox”, the precise details of how the outbox works being of no concern to S. Of course, inside S’s outbox, we know what happens: an encryption ci of mi under k is computed, and this is sent out over the wire to R. On the receiving end, when a ciphertext cˆ is received at R’s end of the wire, it is decrypted using k, and if the decryption is a message m ˆ 6= reject, the message m ˆ is placed in R’s “inbox”. Whenever a message appears in his inbox, R can retrieve it and processes it according to the logic of his protocol, without worrying about how the message got there. An attacker may try to subvert communication between S and R in a number of ways. • First, the attacker may drop, reorder, or duplicate the ciphertexts sent by S. • Second, the attacker may modify ciphertexts sent by S, or inject ciphertexts created out of “whole cloth”. • Third, the attacker may have partial knowledge of some of the messages sent by S, or may even be able to influence the choice of some of these messages. • Fourth, by observing R’s behavior, the attacker may be able to glean partial knowledge of some of the messages processed by R. Even the knowledge of whether or not a ciphertext delivered to R was rejected could be useful. Having described an abstract encryption interface and its implementation, we now describe an ideal implementation of this interface that captures in an intuitive way the guarantees ensured by authenticated encryption. When S drops mi in its outbox, instead of encrypting mi , the ideal implementation creates a ciphertext ci by encrypting a dummy message dummy i , that has nothing to do with mi (except that it should be of the same length). Thus, ci serves as a “handle” for mi , but does not contain any information about mi (other than its length). When ci arrives at R, the corresponding message mi is magically copied from S’s outbox to R’s inbox. If a ciphertext arrives at R that is not among the previously generated ci ’s, the ideal implementation simply discards it. This ideal implementation is just a thought experiment. It obviously cannot be physically realized in any efficient way (without first inventing teleportation). As we shall argue, however, if the underlying cipher E provides authenticated encryption, the ideal implementation is — for all practical purposes — equivalent to the real implementation. Therefore, a protocol designer need not worry about any of the details of the real implementation or the nuances of cryptographic definitions: he can simply pretend he is using the abstract encryption interface with its ideal implementation, in which ciphertexts are just handles and messages magically jump from S to R.
346
Hopefully, analyzing the security properties of the higherlevel protocol will be much easier in this setting. Note that even in the ideal implementation, the attacker may still drop, reorder, or duplicate ciphertexts, and these will cause the corresponding messages to be dropped, reordered, or duplicated. Using sequence numbers and bu↵ers, it is not hard to deal with these possibilities, but that is left to the higherlevel protocol. We now argue informally that when E provides authenticated encryption, the real world implementation is indistinguishable from the ideal implementation. The argument proceeds in three steps. We start with the real implementation, and in each step, we make a slight modification. • First, we modify the real implementation of R’s inbox, as follows. When a ciphertext cˆ arrives on R’s end, the list of ciphertexts c1 , c2 , . . . previously generated by S is scanned, and if cˆ = ci , then the corresponding message mi is magically copied from S’s outbox into R’s inbox, without actually running the decryption algorithm. The correctness property of E ensures that this modification behaves exactly the same as the real implementation. • Second, we modify the implementation on R’s inbox again, so that if a ciphertext cˆ arrives on R’s end that is not among the ciphertexts generated by S, the implementation simply discards cˆ. The only way the adversary could distinguish this modification from the first is if he could create a ciphertext that would not be rejected and was not generated by S. But this is not possible, since E has ciphertext integrity. • Third, we modify the implementation of S’s outbox, replacing the encryption of mi with the encryption of dummy i . The implementation of R’s inbox remains as in the second modification. Note that the decryption algorithm is never used in either the second or third modifications. Therefore, an adversary who can distinguish this modification from the second can be used to directly break the CPAsecurity of E. Hence, since E is CPAsecure, the two modifications are indistinguishable. Since the third modification is identical to the ideal implementation, we see that the real and ideal implementations are indistinguishable from the adversary’s point of view. A technical point we have not considered is the possibility that the ci ’s generated by S are not unique. Certainly, if we are going to view the ci ’s as handles in the ideal implementation, uniqueness would seem to be an essential property. In fact, CPAsecurity implies that the ci ’s generated in the ideal implementation are unique with overwhelming probability — see Exercise 5.11.
9.4
Authenticated encryption ciphers from generic composition
We now turn to constructing authenticated encryption by combining a CPAsecure cipher and a secure MAC. We show that encryptthenMAC is always AEsecure, but MACthenencrypt is not.
347
9.4.1
EncryptthenMAC
Let E = (E, D) be a cipher defined over (Ke , M, C) and let I = (S, V ) be a MAC defined over (Km , C, T ). The encryptthenMAC system EEtM = (EEtM , DEtM ), or EtM for short, is defined as follows: EEtM ( (ke , km ), m)
:=
c R E(ke , m), Output (c, t)
DEtM ((ke , km ), (c, t) )
:=
if V (km , c, t) = reject then output reject otherwise, output D(ke , c)
t
R
S(km , c)
The EtM system is defined over (Ke ⇥ Km , M, C ⇥ T ). The following theorem shows that EEtM provides authenticated encryption. Theorem 9.2. Let E = (E, D) be a cipher and let I = (S, V ) be a MAC system. Then EEtM is AEsecure assuming E is CPAsecure and I is a secure MAC system. Also, EEtM is 1AEsecure assuming E is semantically secure and I is a onetime secure MAC system. In particular, for every ciphertext integrity adversary Aci that attacks EEtM as in Attack Game 9.1 there exists a MAC adversary Bmac that attacks I as in Attack Game 6.1, where Bmac is an elementary wrapper around Aci , and which makes no more signing queries than Aci makes encryption queries, such that CIadv[Aci , EEtM ] = MACadv[Bmac , I]. For every CPA adversary Acpa that attacks EEtM as in Attack Game 5.2 there exists a CPA adversary Bcpa that attacks E as in Attack Game 5.2, where Bcpa is an elementary wrapper around Acpa , and which makes no more encryption queries than does Acpa , such that CPAadv[Acpa , EEtM ] = CPAadv[Bcpa , E].
Proof. Let us first show that EEtM provides ciphertext integrity. The proof is by a straight forward reduction. Suppose Aci is a ciphertext integrity adversary attacking EEtM . We construct a MAC adversary Bmac attacking I. Adversary Bmac plays the role of adversary in a MAC attack game for I. It interacts with a MAC challenger Cmac that starts by picking a random km R Km . Adversary Bmac works by emulating a EEtM ciphertext integrity challenger for Aci , as follows: ke R K e upon receiving a query mi 2 M from Aci do: ci R E(ke , mi ) Query Cmac on ci and obtain ti R S(km , ci ) in response Send (ci , ti ) to Aci // then (ci , ti ) = EEtM ( (ke , km ), mi )
eventually Aci outputs a ciphertext (c, t) 2 C ⇥ T output the messagetag pair (c, t)
It should be clear that Bmac responds to Aci ’s queries as in a real ciphertext integrity attack game. Therefore, with probability CIadv[Aci , EEtM ] adversary Aci outputs a ciphertext (c, t) that makes it win Attack Game 9.1 so that (c, t) 62 {(c1 , t1 ), . . .} and V (km , c, t) = accept. It follows that (c, t) 348
is a messagetag pair that lets Bmac win the MAC attack game and therefore CIadv[Aci , EEtM ] = MACadv[Bmac , I], as required. It remains to show that if E is CPAsecure then so is EEtM . This simply says that the tag included in the ciphertext, which is computed using the key km (and does not involve the encryption key ke at all), does not help the attacker break CPA security of EEtM . This is straightforward and is left as an easy exercise (see Exercise 5.20). 2 Recall that our definition of a secure MAC from Chapter 6 requires that given a messagetag pair (c, t) the attacker cannot come up with a new tag t0 6= t such that (c, t0 ) is a valid messagetag pair. At the time it seemed odd to require this: if the attacker already has a valid tag for c, why do we care if he finds another tag for c? Here we see that if the attacker could come with a new valid tag t0 for c then he could break ciphertext integrity for EtM. From an EtM ciphertext (c, t) the attacker could construct a new valid ciphertext (c, t0 ) and win the ciphertext integrity game. Our definition of secure MAC ensures that the attacker cannot modify an EtM ciphertext without being detected. Common mistakes in implementing encryptthenMAC A common mistake when implementing encryptthenMAC is to use the same key for the cipher and the MAC, i.e., setting ke = km . The resulting system need not provide authenticated encryption and can be insecure, as shown in Exercise 9.8. In the proof of Theorem 9.2 we relied on the fact that the two keys ke and km are chosen independently. Another common mistake is to apply the MAC signing algorithm to only part of the ciphertext. We look at an example. Suppose the underlying CPAsecure cipher E = (E, D) is randomized CBC mode (Section 5.4.3) so that the encryption of a message m is (r, c) R E(k, c) where r is a random IV. When implementing encryptthenMAC EEtM = (EEtM , DEtM ) the encryption algorithm is incorrectly defined as EEtM (ke , km ), m :=
(r, c)
R
E(ke , m), t
R
S(km , c), output (r, c, t) .
Here, E(ke , m) outputs the ciphertext (r, c), but the MAC signing algorithm is only applied to c; the IV is not protected by the MAC. This mistake completely destroys ciphertext integrity: given a ciphertext (r, c, t) an attacker can create a new valid ciphertext (r0 , c, t) for some r0 6= r. The decryption algorithm will not detect this modification of the IV and will not output reject. Instead, the decryption algorithm will output D ke , (r0 , c) . Since (r0 , c, t) is a valid ciphertext the adversary wins the ciphertext integrity game. Even worse, if (r, c, t) is the encryption of a message m then changing (r, c, t) to (r , c, t) for any causes the CBC decryption algorithm 0 0 to output a message m where m [0] = m[0] . This means that the attacker can change header information in the first block of m to any value of the attacker’s choosing. An early edition of the ISO 19772 standard for authenticated encryption made precisely this mistake [81]. Similarly, in 2013 it was discovered that the RNCryptor facility in Apple’s iOS, built for data encryption, used a faulty encryptthenMAC where the HMAC was not applied to the encryption IV [84]. Another pitfall to watch out for in an implementation is that no plaintext data should be output before the integrity tag over the entire message is verified. See Section 9.9 for an example of this.
349
9.4.2
MACthenencrypt is not generally secure: padding oracle attacks on SSL
Next, we consider the MACthenencrypt generic composition of a CPA secure cipher and a secure MAC. We show that this construction need not be AEsecure and can lead to many realworld problems. To define MACthenencrypt precisely, let I = (S, V ) be a MAC defined over (Km , M, T ) and let E = (E, D) be a cipher defined over (Ke , M ⇥ T , C). The MACthenencrypt system EMtE = (EMtE , DMtE ), or MtE for short, is defined as follows: EMtE ( (ke , km ), m)
:=
t R S(km , m), Output c
DEtM ((ke , km ), c )
:=
(m, t) D(ke , c) if V (km , m, t) = reject then output reject otherwise, output m
c
R
E(kc , (m, t) )
The MtE system is defined over (Ke ⇥ Km , M, C). A badly broken MtE cipher. We show that MtE is not guaranteed to be AEsecure even if E is a CPAsecure cipher and I is a secure MAC. In fact, MtE can fail to be secure for widelyused ciphers and MACs and this has lead to many significant attacks on deployed systems. Consider the SSL 3.0 protocol used to protect WWW traffic for over two decades (the protocol is disabled in modern browsers). SSL 3.0 uses MtE to combine randomized CBC mode encryption and a secure MAC. We showed in Chapter 5 that randomized CBC mode encryption is CPAsecure, yet this combination is badly broken: an attacker can e↵ectively decrypt all traffic using a chosen ciphertext attack. This leads to a devastating attack on SSL 3.0 called POODLE [18]. Let us assume that the underlying block cipher used in CBC operates on 16 byte blocks, as in AES. Recall that CBC mode encryption pads its input to a multiple of the block length and SSL 3.0 does so as follows: if a pad of length p > 0 bytes is needed, the scheme pads the message with p 1 arbitrary bytes and adds one additional byte whose value is set to (p 1). If the message length is already a multiple of the block length (16 bytes) then SSL 3.0 adds a dummy block of 16 bytes where the last byte is set to 15 and the first 15 bytes are arbitrary. During decryption the pad is removed by reading the last byte and removing that many more bytes. Concretely, the cipher EMtE = (EMtE , DMtE ) obtained from applying MtE to randomized CBC mode encryption and a secure MAC works as follows: • EMtE ( (ke , km ), m): First use the MAC signing algorithm to compute a fixedlength tag t R S(km , m) for m. Next, encrypt m k t with randomized CBC encryption: pad the message and then encrypt in CBC mode using key ke and a random IV. Thus, the following data is encrypted to generate the ciphertext c: message m
tag t
pad p
(9.8)
Notice that the tag t does not protect the integrity of the pad. We will exploit this to break CPA security using a chosen ciphertext attack. • DMtE ( (ke , km ), c): Run CBC decryption to obtain the plaintext data in (9.8). Next, remove the pad p by reading the last byte in (9.8) and removing that many more bytes from the data (i.e., if the last byte is 3 then that byte is removed plus 3 additional bytes). Next, verify the MAC tag and if valid return the remaining bytes as the message. Otherwise, output reject. 350
Both SSL 3.0 and TLS 1.0 use a defective variant of randomized CBC encryption, discussed in Exercise 5.12, but this is not relevant to our discussion here. Here we will assume that a correct implementation of randomized CBC encryption is used. The chosen ciphertext attack. We show a chosen ciphertext attack on the system EMtE that lets the adversary decrypt any ciphertext of its choice. It follows that EMtE need not be AEsecure, even though the underlying cipher is CPAsecure. Throughout this section we let (E, D) denote the block cipher used in CBC mode encryption. It operates on 16byte blocks. Suppose the adversary intercepts a valid ciphertext c := EMtE ( (ke , km ), m) for some unknown message m. The length of m is such that after a MAC tag t is appended to m the length of (m k t) is a multiple of 16 bytes. This means that a full padding block of 16 bytes is appended during CBC encryption and the last byte of this pad is 15. Then the ciphertext c looks as follows: c
=

c[0] {z IV
}
c[1]
···
{z
encryption of m
}
c[` {z
1]
encrypted tag
}
c[`] {z

c[1] {z }
}
encrypted pad
Lets us first show that the adversary can learn something about m[0] (the first 16byte block of m). This will break semantic security of EMtE . The attacker prepares a chosen ciphertext query cˆ by replacing the last block of c with c[1]. That is, :=
cˆ
c[0]
···
c[1]
c[`
1]
(9.9)
encrypted pad?
By definition of CBC decryption, decrypting the last block of cˆ yields the 16byte plaintext block v := D ke , c[1]
c[`
1] = m[0]
c[0]
c[`
1].
If the last byte of v is 15 then during decryption the entire last block will be treated as a padding block and removed. The remaining string is a valid messagetag pair and will decrypt properly. If the last byte of v is not 15 then most likely the response to the decryption query will be reject. Put another way, if the response to a decryption query for cˆ is not reject then the attacker learns that the last byte of m[0] is equal to the last byte of u := 15 c[0] c[` 1]. Otherwise, the attacker learns that the last byte of m[0] is not equal to the last byte of u. This directly breaks semantic security of the EMtE : the attacker learned something about the plaintext m. We leave it as an instructive exercise to recast this attack in terms of an adversary in a chosen ciphertext attack game (as in Attack Game 9.2). With a single plaintext query followed by a single ciphertext query the adversary has advantage 1/256 in winning the game. This already proves that EMtE is insecure. Now, suppose the attacker obtains another encryption of m, call it c0 , using a di↵erent IV. The attacker can use the ciphertexts c and c0 to form four useful chosen ciphertext queries: it can replace the last block of either c or c0 with either of c[1] or c0 [1]. By issuing these four ciphertext queries the attacker learns if the last byte of m[0] is equal to the last byte of one of 15
c[0]
c[`
1],
15
c[0]
c0 [`
1],
15
c0 [0]
c[`
1],
15
c0 [0]
c0 [`
1].
If these four values are distinct they give the attacker four chances to learn the last byte of m[0]. Repeating this multiple times with more fresh encryptions of the message m will quickly reveal the 351
last byte of m[0]. Each chosen ciphertext query reveals that byte with probability 1/256. Therefore, on average, with 256 chosen ciphertext queries the attacker learns the exact value of the last byte of m[0]. So, not only can the attacker break semantic security, the attacker can actually recover one byte of the plaintext. Next, suppose the adversary could request an encryption of m shifted one byte to the right to obtain a ciphertext c1 . Plugging c1 [1] into the last block of the ciphertexts from the previous phase (i.e., encryptions of the unshifted m) and issuing the resulting chosen ciphertext queries reveals the second to last byte of m[0]. Repeating this for every byte of m eventually reveals all of m. We show next that this gives a real attack on SSL 3.0. A complete break of SSL 3.0. Chosen ciphertext attacks may seem theoretical, but they frequently translate to devastating realworld attacks. Consider a Web browser and a victim Web server called bank.com. The two exchange information encrypted using SSL 3.0. The browser and server have a shared secret called a cookie and the browser embeds this cookie in every request that it sends to bank.com. That is, abstractly, requests from the browser to bank.com look like: GET path cookie: cookie where path identifies the name of a resource being requested from bank.com. The browser only inserts the cookie into requests it sends to bank.com The attacker’s goal is to recover the secret cookie. First it makes the browser visit attacker.com where it sends a Javascript program to the browser. This Javascript program makes the browser issue a request for resource “/AA” at bank.com. The reason for this particular path is to ensure that the length of the message and MAC is a multiple of the block size (16 bytes), as needed for the attack. Consequently, the browser sends the following request to bank.com GET /AA
cookie: cookie
(9.10)
encrypted using SSL 3.0. The attacker can intercept this encrypted request c and mounts the chosen ciphertext attack on MtE to learn one byte of the cookie. That is, the attacker prepares cˆ as in (9.9), sends cˆ to bank.com and looks to see if bank.com responds with an SSL error message. If no error message is generated then the attacker learns one byte of the cookie. The Javascript can cause the browser to repeatedly issue the request (9.10) giving the adversary the fresh encryptions needed to eventually learn one byte of the cookie. Once the adversary learns one byte of the cookie it can shift the cookie one byte to the right by making the Javascript program issue a request to bank.com for GET /AAA
cookie: cookie
This gives the attacker a block of ciphertext, call it c1 [2], where the cookie is shifted one byte to the right. Resending the requests from the previous phase to the server, but now with the last block replaced by c1 [2], eventually reveals the second byte of the cookie. Iterating this process for every byte of the cookie eventually reveals the entire cookie. In e↵ect, Javascript in the browser provides the attacker with the means to mount the desired chosen plaintext attack. Intercepting packets in the network, modifying them and observing the server’s response, gives the attacker the means to mount the desired chosen ciphertext attack. The combination of these two completely breaks MtE encryption in SSL 3.0. 352
One minor detail is that whenever bank.com responds with an SSL error message the SSL session shuts down. This does not pose a problem: every request that the Javascript running in the browser makes to bank.com initiates a new SSL session. Hence, every chosen ciphertext query is encrypted under a di↵erent session key, but that makes no di↵erence to the attack: every query tests if one byte of the cookie is equal to one known random byte. With enough queries the attacker learns the entire cookie.
9.4.3
More padding oracle attacks.
TLS 1.0 is an updated version of SSL 3.0. It defends against the attack of the previous section by adding structure to the pad as explained in Section 5.4.4: when padding with p bytes, all bytes of the pad are set to p 1. Moreover, during decryption, the decryptor is required to check that all padding bytes have the correct value and reject the ciphertext if not. This makes it harder to mount the attack of the previous section. Of course our goal was merely to show that MtE is not generally secure and SSL 3.0 made that abundantly clear. A padding oracle timing attack. Despite the defenses in TLS 1.0 a naive implementation of MtE decryption may still be vulnerable. Suppose the implementation works as follows: first it applies CBC decryption to the received ciphertext; next it checks that the pad structure is valid and if not it rejects the ciphertext; if the pad is valid it checks the integrity tag and if valid it returns the plaintext. In this implementations the integrity tag is checked only if the pad structure is valid. This means that a ciphertext with an invalid pad structure is rejected faster than a ciphertext with a valid pad structure, but an invalid tag. An attacker can measure the time that the server takes to respond to a chosen ciphertext query and if a TLS error message is generated quickly it learns that the pad structure was invalid. Otherwise, it learns that the pad structure was valid. This timing channel is called a padding oracle sidechannel. It is a good exercise to devise a chosen ciphertext attack based on this behavior to completely decrypt a secret cookie, as we did for SSL 3.0. To see how this might work, suppose an attacker intercepts an encrypted TLS 1.0 record c. Let m be the decryption of c. Say the attacker wishes to test if the last byte of m[2] is equal to some fixed byte value b. Let B be an arbitrary 16byte block whose last byte is b. The attacker creates a new ciphertext block cˆ[1] := c[1] B and sends the 3block record cˆ = (c[0], cˆ[1], c[2]) to the server. After CBC decryption of cˆ, the last plaintext block will be := cˆ[1] m[2] ˆ
D(k, c[2]) = m[2]
B.
If the last byte of m[2] is equal to b then m[2] ˆ ends in zero which is a valid pad. The server will attempt to verify the integrity tag resulting in a slow response. If the last byte of m[2] is not equal to b then m[2] ˆ will not end in 0 and will likely end in an invalid pad, resulting in a fast response. By measuring the response time the attacker learns if the last byte of m[2] is equal to b. Repeating this with many chosen ciphertext queries, as we did for SSL 3.0, reveals the entire secret cookie. An even more sophisticated padding oracle timing attack on MtE, as used in TLS 1.0, is called Lucky13 [3]. It is quite challenging to implement TLS 1.0 decryption in way that hides the timing information exploited by the Lucky13 attack. Informative error messages. To make matters worse, the TLS 1.0 specification [31] states that the server should send one type of error message (called bad record mac) when a received 353
ciphertext is rejected because of a MAC verification error and another type of error message (decryption failed) when the ciphertext is rejected because of an invalid padding block. In principle, this tells the attacker if a ciphertext was rejected because of an invalid padding block or because of a bad integrity tag. This could have enabled the chosen ciphertext attack of the previous paragraph without needing to resort to timing measurements. Fortunately, the error messages are encrypted and the attacker cannot see the error code. Nevertheless, there is an important lesson to be learned here: when decryption fails, the system should never explain why. A generic ‘decryption failed’ code should be sent without o↵ering any other information. This issue was recognized and addressed in TLS 1.1. Moreover, upon decryption failure, a correct implementation should always take the same amount of time to respond, no matter the failure reason.
9.4.4
Secure instances of MACthenencrypt
Although MtE is not generally secure when applied to a CPAsecure cipher, it can be shown to be secure for specific CPA ciphers discussed in Chapter 5. We show in Theorem 9.3 below that if E happens to implement randomized counter mode, then MtE is secure. In Exercise 9.9 we show that the same holds for randomized CBC, assuming there is no message padding. Theorem 9.3 shows that MACthenencrypt with randomized counter mode is AEsecure even if the MAC is only onetime secure. That is, it suffices to use a weak MAC that is only secure against an adversary that makes a single chosen message query. Intuitively, the reason we can prove security using such a weak MAC is that the MAC value is encrypted, and consequently it is harder for the adversary to attack the MAC. Since onetime MACs are a little shorter and faster than manytime MACs, MACthenencrypt with randomized counter mode has a small advantage over encryptthenMAC. Nevertheless, the attacks on MACthenencrypt presented in the previous section suggest that it is difficult to implement correctly, and should not be used. Our starting point is a randomized countermode cipher E = (E, D), as discussed in Section 5.4.2. We will assume that E has the general structure as presented in the case study on AES counter mode at the end of Section 5.4.2 (page 189). Namely, we use a countermode variant where the cipher E is built from a secure PRF F defined over (Ke , X ⇥ Z` , Y), where Y := {0, 1}n . More precisely, for a message m 2 Y ` algorithm E works as follows: 9 8 > > x R X > > > > = < for j = 0 to m 1: := E(ke , m) u[j] F ke , (x, j) m[j] > > > > > > ; : output c := (x, u) 2 X ⇥ Y m
Algorithm D(ke , c) is defined similarly. Let I = (S, V ) be a secure onetime MAC defined over (Km , M, T ) where M := Y `m and T := Y `t , and where `m + `t < `. The MACthenencrypt cipher EMtE = (EMtE , DMtE ), built from F and I and taking messages in M, is defined as follows: EMtE (ke , km ), m := t R S(km , m), c R E ke , (m k t) , output c 9 8 D(ke , c) > > = < (m k t) DMtE (ke , km ), c := if V (km , m, t) = reject then output reject > > ; : otherwise, output m 354
(9.11)
As we discussed at the end of Section 9.4.1, and in Exercise 9.8, the two keys ke and km must be chosen independently. Setting ke = km will invalidate the following security theorem. Theorem 9.3. The cipher EMtE = (EMtE , DMtE ) in (9.11) built from the PRF F and MAC I provides authenticated encryption assuming I is a secure onetime MAC and F is a secure PRF where 1/X  is negligible. In particular, for every Qquery ciphertext integrity adversary Aci that attacks EMtE as in Attack 0 that attack I as in Attack Game 6.1, Game 9.1 there exists two MAC adversaries Bmac and Bmac and a PRF adversary Bprf that attacks F as in Attack Game 4.2, each of which is an elementary wrapper around Aci , such that CIadv[Aci , EMtE ] PRFadv[Bprf , F ] + 0 , I] + Q · MAC1 adv[Bmac , I] + MAC1 adv[Bmac
Q2 . 2X 
(9.12)
For every CPA adversary Acpa that attacks EEtM as in Attack Game 5.2 there exists a CPA adversary Bcpa that attacks E as in Attack Game 5.2, which is an elementary wrapper around Acpa , such that CPAadv[Acpa , EMtE ] = CPAadv[Bcpa , E]
Proof idea. CPA security of the system follows immediately from CPA security of randomized counter mode. The challenge is to prove ciphertext integrity for EMtE . So let Aci be a ciphertext integrity adversary. This adversary makes a series of queries, m1 , . . . , mQ . For each mi , the CI challenger gives to Aci a ciphertext ci = (xi , ui ), where xi is a random IV, and ui is a onetime pad encryption of the pair mi k ti using a pseudorandom pad ri derived from xi using the PRF F . Here, ti is a MAC tag computed on mi . At the end of the attack game, adversary Aci outputs a ciphertext c = (x, u), which is not among the ci ’s, and wins if c is a valid ciphertext. This means that u decrypts to m k t using a pseudorandom pad r derived from x, and t is a valid tag on m. Now, using the PRF security property and the fact that the xi ’s are unlikely to repeat, we can e↵ectively replace the pseudorandom ri ’s (and r) with truly random pads, without a↵ecting Aci ’s advantage significantly. This is where the terms PRFadv[Bprf , F ] and Q2 /2X  in (9.12) come from. Note that after making this modification, the ti ’s are perfectly hidden from the adversary. We then consider two di↵erent ways in which Aci can win in this modified attack game. • In the first way, the value x output by Aci is not among the xi ’s. But in this case, the only way for Aci to win is to hope that a random tag on a random message is valid. This is where 0 , I] in (9.12) comes from. the term MAC1 adv[Bmac • In the second way, the value x is equal to xj for some j = 1, . . . , Q. In this case, to win, the value u must decrypt under the pad rj to m k t where t is a valid tag on m. Moreover, since c 6= cj , we have (m, t) 6= (mj , tj ). To turn Aci into a onetime MAC adversary, we have to guess the index j in advance: for all indices i di↵erent from the guessed index, we can replace the tag ti by a dummy tag. This guessing strategy is where the term Q · MAC1 adv[Bmac , I] in (9.12) comes from. 2 Proof. To prove ciphertext integrity, we let Aci interact with a number of closely related challengers. For j = 0, 1, 2, 3, 4 we define Wj to be the event that the adversary wins in Game j. Game 0. As usual, we begin by letting Aci interact with the standard ciphertext integrity challenger in Attack Game 9.1 as it applies to EMtE , so that Pr[W0 ] = CIadv[Aci , EMtE ]. 355
Game 1. Now, we replace the pseudorandom pads in the countermode cipher by truly independent onetime pads. Since F is a secure PRF and 1/X  is negligible, the adversary will not notice the di↵erence. The resulting CI challenger for EMtE works as follows. km R K m // Choose random MAC key R ! {1, . . . , Q} // this ! will be used in Game 3 upon receiving the ith query mi 2 Y `m for i = 1, 2, . . . do: S(km , mi ) 2 T // compute the tag for mi (1) ti R (2) xi X // Choose a random IV
ri R Y mi +`t // Choose a sufficiently long truly random onetime pad ui (mi k ti ) ri , ci (xi , ui ) // build ciphertext send ci to the adversary
At the end of the game, Aci outputs c = (x, u), which is not among c1 , . . . , cQ , and the winning condition is evaluated as follows: (3) (4)
// decrypt ciphertext c if x = xj for some j then (m k t) otherwise, r R Y u and (m k t) Aci wins if V (km , m, t) = accept
u rj u r //
check resulting messagetag pair
Note that for specificity, in line (3) if there is more than one j for which x = xj , we can take the smallest such j. A standard argument shows that there exists an efficient PRF adversary Bprf such that: Pr[W1 ]
Pr[W0 ] PRFadv[Bprf , F ] +
Q2 . 2X 
(9.13)
Note that if we wanted to be a bit more careful, we would break this argument up into two steps. In the first step, we would play our “PRF card” to replace F (ke , ·) be a truly random function f . This introduces the term PRFadv[Bprf , F ] in (9.13). In the second step, we would use the “forgetful gnome” technique to make all the outputs of f independent. Using the Di↵erence Lemma applied to the event that all of the xi ’s are distinct introduces the term Q2 /2X  in (9.13). Game 2. Now we restrict the adversary’s winning condition to require that the IV used in the final ciphertext c is the same as one of the IVs given to Aci during the game. In particular, we replace line (4) with (4)
otherwise, the adversary loses in Game 2.
Let Z2 be the event that in Game 2, the final ciphertext c = (x, u) from Aci is valid despite using a previously unused x 2 X . We know that the two games proceed identically, unless event Z2 happens. When event Z2 happens in Game 2 then the resulting pair (m, t) is uniformly random in Y u `t ⇥ Y `t . Such a pair is unlikely to form a valid messagetag pair. Not only that, the challenger in Game 2 e↵ectively encrypts all of the tags ti generated in line (1) with a onetime pad, so these tags could be replaced by dummy tags, without a↵ecting the probability that Z2 0 occurs. Based on these observations, we can easily construct an efficient MAC adversary Bmac such 0 0 that Pr[Z2 ] MAC1 adv[Bmac , I]. Adversary Bmac runs as follows. It plays the role of challenger to Aci as in Game 2, except that in line (1) above, it computes ti 0`t . When Aci outputs c = (x, u), 356
0 adversary Bmac generates outputs a random pair in Y u we have
Pr[W2 ]
`t
⇥ Y `t . Hence, by the di↵erence lemma,
0 Pr[W1 ] MAC1 adv[Bmac , I].
(9.14)
Game 3. We further constrain the adversary’s winning condition by requiring that the ciphertext forgery use the IV from ciphertext number ! given to Aci . Here ! is a random number in {1, . . . , Q} chosen by the challenger. The only change to the winning condition of Game 2 is that line (3) now becomes: (3) (4)
u r! if x = x! then (m k t) otherwise, the adversary loses in Game 2.
Since ! is independent of Aci ’s view, we know that Pr[W3 ]
(1/Q) · Pr[W2 ]
(9.15)
Game 4. Finally, we change the challenger so that it only computes a valid tag for query number ! issued by Aci . For all other queries the challenger just makes up an arbitrary (invalid) tag. Since the tags are encrypted using onetime pads the adversary cannot tell that he is given encryptions of invalid tags. In particular, the only di↵erence from Game 3 is that we replace line (1) by the following two lines: (1)
ti (0n )`t 2 T if i = ! then ti
S(km , mi ) 2 T
//
only compute correct tag for m!
Since the adversary’s view in this game is identical to its view in Game 3 we have Pr[W4 ] = Pr[W3 ]
(9.16)
Final reduction. We claim that there is an efficient onetime MAC forger Bmac so that Pr[W4 ] = MAC1 adv[Bmac , I]
(9.17)
Adversary Bmac interacts with a MAC challenger C and works as follows: ! R {1, . . . , Q} upon receiving the ith query mi 2 {0, 1}`m for i = 1, 2, . . . do: (0n )`t 2 T ti if i = ! then query C for the tag on mi and let ti 2 T be the response // Choose a random IV xi R X // Choose a sufficiently long random onetime pad ri R Y m+`t (mi k ti ) ri , ci (xi , ui ) ui send ci to the adversary when Aci outputs c = (x, u) from Aci do: if x = x! then (m k t) u r! output (m, t) as the messagetag forgery Since c 6= c! we know that (m, t) 6= (m! , t! ). Hence, whenever Aci wins Game 4 we know that Bmac does not abort, and outputs a pair (m, t) that lets it win the onetime MAC attack game. It follows that Pr[W4 ] = MAC1 adv[Bmac , I] as required. In summary, putting equations (9.13)–(9.17) together proves the theorem. 2 357
9.4.5
EncryptthenMAC or MACthenencrypt?
So far we proved the following facts about the MtE and EtM modes: • EtM provides authenticated encryption whenever the cipher is CPAsecure and the MAC is secure. The MAC on the ciphertext prevents any tampering with the ciphertext. • MtE is not generally secure — there are examples of CPAsecure ciphers for which the MtE system does is not AEsecure. Moreover, MtE is difficult to implement correctly due to a potential timing sidechannel that leads to serious chosen ciphertext attacks. However, for specific ciphers, such as randomized counter mode and randomized CBC, the MtE mode is AEsecure even if the MAC is only onetime secure. • A third mode, called encryptandMAC (EaM), is discussed in Exercise 9.10. The exercise shows that EaM is secure when using randomized countermode cipher as long as the MAC is a secure PRF. EaM is inferior to EtM in every respect and should not be used. These facts, and the example attacks on MtE, suggest that EtM is the better mode to use. Of course, it is critically important that the underlying cipher be CPAsecure and the underlying MAC be a secure MAC. Otherwise, EtM may provide no security at all. Given all the past mistakes in implementing these modes it is advisable that developers not implement EtM themselves. Instead, it is best to use an encryption standard, like GCM (see Section 9.7), that uses EtM to provide authenticated encryption out of the box.
9.5
Noncebased authenticated encryption with associated data
In this section we extend the syntax of authenticated encryption to match the way in which it is commonly used. First, as we did for encryption and for MACs, we define noncebased authenticated encryption where we make the encryption and decryption algorithms deterministic, but let them take as input a unique nonce. This approach can reduce ciphertext size and also improve security. Second, we extend the encryption algorithm by giving it an additional input message, called associated data, whose integrity is protected by the ciphertext, but its secrecy is not. The need for associated data comes up in a number of settings. For example, when encrypting packets in a networking protocol, authenticated encryption protects the packet body, but the header must be transmitted in the clear so that the network can route the packet to its intended destination. Nevertheless, we want to ensure header integrity. The header is provided as the associated data input to the encryption algorithm. A cipher that supports associated data is called an AD cipher. The syntax for a noncebased AD cipher E = (E, D) is as follows: c = E(k, m, d, N ), where c 2 C is the ciphertext, k 2 K is the key, m 2 M is the message, d 2 D is the associated data, and N 2 N is the nonce. Moreover, the encryption algorithm E is required to be deterministic. Likewise, the decryption syntax becomes D(k, c, d, N ) which outputs a message m or reject. We say that the noncebased AD cipher is defined over (K, M, D, C, N ). As usual, we require that ciphertexts generated by E are correctly decrypted 358
by D, as long as both are given the same nonce and associated data. That is, for all keys k, all messages m, all associated data d, and all nonces N 2 N : D k, E(k, m, d,
N ),
d,
N
= m.
If the message m given as input to the encryption algorithm is the empty message then cipher (E, D) essentially becomes a MAC system for the associated data d. CPA security. A noncebased AD cipher is CPAsecure if it does not leak any useful information to an eavesdropper assuming that no nonce is used more than once in the encryption process. CPA security for a noncebased AD cipher is defined as CPA security for a standard noncebased cipher (Section 5.5). The only di↵erence is in the encryption queries. Encryption queries in Experiment b, for b = 0, 1, are processed as follows: The ith encryption query is a pair of messages, mi0 , mi1 2 M, of the same length, associated data di 2 D, and a unique nonce N i 2 N \ {N 1 , . . . , N i 1 }.
The challenger computes ci
E(k, mib , di , N i ), and sends ci to the adversary.
Nothing else changes from the definition in Section 5.5. Note that the associated data di is under the adversary’s control, as are the nonces N i , subject to the nonces being unique. For b = 0, 1, let Wb be the event that A outputs 1 in Experiment b. We define A’s advantage with respect to E as nCPAad adv[A, E] := Pr[W0 ]
Pr[W1 ].
2
Definition 9.7 (CPA security). A noncebased AD cipher is called semantically secure against chosen plaintext attack, or simply CPAsecure, if for all efficient adversaries A, the quantity nCPAad adv[A, E] is negligible. Ciphertext integrity. A noncebased AD cipher provides ciphertext integrity if an attacker who can request encryptions under key k for messages, associated data, and nonces of his choice cannot output a new triple (c, d, N ) that is accepted by the decryption algorithm. The adversary, however, must never issue an encryption query using a previously used nonce. More precisely, we modify the ciphertext integrity game (Attack Game 9.1) as follows: Attack Game 9.3 (ciphertext integrity). For a given AD cipher E = (E, D) defined over (K, M, D, C, N ), and a given adversary A, the attack game runs as follows: • The challenger chooses a random k
R
K.
• A queries the challenger several times. For i = 1, 2, . . . , the ith query consists of a message mi 2 M, associated data di 2 D, and a previously unused nonce R N i 2 N \ {N 1 , . . . , N i 1 }. The challenger computes ci E(k, mi , di , N i ), and gives ci to A. • Eventually A outputs a candidate triple (c, d, N ) where c 2 C, d 2 D, and that is not among the triples it was given, i.e., (c, d, N ) 62 {(c1 , d1 , N 1 ), (c2 , d2 , N 2 ), . . .}. 359
N
2N
We say that A wins the game if D(k, c, d, N ) 6= reject. We define A’s advantage with respect to E, denoted nCIad adv[A, E], as the probability that A wins the game. 2 Definition 9.8. We say that a noncebased AD cipher E = (E, D) has ciphertext integrity if for all efficient adversaries A, the value nCIad adv[A, E] is negligible. Authenticated encryption. We can now define noncebased authenticated encryption for an AD cipher. We refer to this notion as a noncebased AEAD cipher which is shorthand for authenticated encryption with associated data. Definition 9.9. We say that a noncebased AD cipher E = (E, D) provides authenticated encryption, or is simply a noncebased AEAD cipher, if E is CPAsecure and has ciphertext integrity. Generic encryptthenMAC composition. We construct a noncebased AEAD cipher E = (EEtM , DEtM ) by combining a noncebased CPAsecure cipher (E, D) (as in Section 5.5) with a noncebased secure MAC (S, V ) (as in Section 7.5) as follows: EEtM ( (ke , km ), m, d, N )
:=
c E(ke , m, N ), Output (c, t)
DEtM ((ke , km ), (c, t), d, N )
:=
if V (km , (c, d), t, N ) = reject then output reject otherwise, output D(ke , c, d, N )
t
S(km , (c, d), N )
The EtM system is defined over (Ke ⇥ Km , M, D, C ⇥ T , N ). The following theorem shows that EEtM is a secure AEAD cipher. Theorem 9.4. Let E = (E, D) be a noncebased cipher and let I = (S, V ) be a noncebased MAC system. Then EEtM is a noncebased AEAD cipher assuming E is CPAsecure and I is a secure MAC system. The proof of Theorem 9.4 is essentially the same as the proof of Theorem 9.2.
9.6
One more variation: CCAsecure ciphers with associated data
In Section 9.5, we introduced two new features to our ciphers: nonces and associated data. There are two variations we could consider: ciphers with nonces but without associated data, and ciphers with associated data but without nonces. We could also consider all of these variations with respect to other security notions, such as CCA security. Considering all of these variations in detail would be quite tedious. However, we consider one variation that will be important later in the text, namely CCAsecure ciphers with associated data (but without nonces). To define this notion, we begin by defining the syntax for a cipher with associated data, or AD cipher, without nonces. For such a cipher E = (E, D), the encryption algorithm may be probabilistic and works as follows: c R E(k, m, d), where c 2 C is the ciphertext, k 2 K is the key, m 2 M is the message, and d 2 D is the associated data. The decryption syntax is D(k, c, d), 360
which outputs a message m or reject. We say that the AD cipher is defined over (K, M, D, C). As usual, we require that ciphertexts generated by E are correctly decrypted by D, as long as both are given the same associated data. That is, ⇥ ⇤ Pr D k, E(k, m, d), d = m = 1.
Definition 9.10 (CCA and 1CCA security with associated data). The definition of CCA security for ordinary ciphers carries over naturally to AD ciphers. Attack Game 9.2 is modified as follows. For encryption queries, in addition to a pair of messages (mi0 , mi1 ), the adversary also submits associated data di , and the challenger computes ci R E(k, mib , di ). For decryption queries, in addition to a ciphertext cˆj , the adversary submits associated data dˆj , and the challenger D(k, cˆj , dˆj ). The restriction is that the pair (ˆ cj , dˆj ) may not be among the pairs computes m ˆj (c1 , d1 ), (c2 , d2 ), . . . corresponding to previous encryption queries. An adversary A’s advantage in this game is denoted CCAad adv[A, E], and the cipher is said to be CCA secure if this advantage is negligible for all efficient adversaries A. If we restrict the adversary to a single encryption query, as in Definition 9.6, the advantage is denoted 1CCAad adv[A, E], and the cipher is said to be 1CCA secure if this advantage is negligible for all efficient adversaries A. Generic encryptthenMAC composition. In later applications, the notion that we will use is 1CCA security, so for simplicity we focus on that notion for now. We construct a 1CCAsecure AD cipher E = (EEtM , DEtM ) by combining a semantically secure cipher (E, D) with a onetime MAC (S, V ) as follows: EEtM ( (ke , km ), m, d)
:=
c R E(ke , m), Output (c, t)
DEtM ((ke , km ), (c, t), d)
:=
if V (km , (c, d), t) = reject then output reject otherwise, output D(ke , c, d)
t
R
S(km , (c, d))
The EtM system is defined over (Ke ⇥ Km , M, D, C ⇥ T ). Theorem 9.5. Let E = (E, D) be a semantically secure cipher and let I = (S, V ) be a onetime secure MAC system. Then EEtM is a 1CCAsecure AD cipher. The proof of Theorem 9.5 is straightforward, and we leave it as an exercise to the reader. We observe that in most common implementations of the semantically secure cipher E = (E, D), the encryption algorithm E is deterministic. Likewise, in the most common implementations of the onetime secure MAC I = (S, V ), the signing algorithm is deterministic. So for such implementations, the resulting 1CCAsecure AD cipher will have a deterministic encryption algorithm.
9.7
Case study: Galois counter mode (GCM)
Galois counter mode (GCM) is a popular noncebased AEAD cipher standardized by NIST in 2007. GCM is an encryptthenMAC cipher combining a CPAsecure cipher and a secure MAC. The CPA secure cipher is noncebased counter mode, usually using AES. The secure MAC is a CarterWegman MAC built from a keyed hash function called GHASH, a variant of the function Hxpoly from Section 7.4. When encrypting the empty message the cipher becomes a MAC system called GMAC providing integrity for the associated data. 361
GCM uses an underlying block cipher E = (E, D) such as AES defined over (K, X ) where X := {0, 1}128 . The block cipher is used for both counter mode encryption and the CarterWegman MAC. The GHASH function is defined over (X , X ` , X ) for ` := 232 1. GCM can take variable size nonces, but let us first describe GCM using a 96bit nonce N which is the simplest case. The GCM encryption algorithm operates as follows: input: key k 2 K, message m, associated data d, and nonce E(k, 0128 )
km
//
N
2 {0, 1}96
first, generate the key for GHASH (a variant of Hxpoly )
Compute the initial value of the counter in counter mode encryption: x (N k 031 1) 2 {0, 1}128 x0 x + 1 // initial value of counter c d0 c0 (⇤)
{encryption of m using counter mode starting the counter at x0 } {pad d with zeros to closest multiple of 128 bits} {pad c with zeros to closest multiple of 128 bits}
Compute the CarterWegman MAC: ⌘ ⇣ 0 2 {0, 1}128 h GHASH km , d k c0 k length(d) k length(c) t
h
output (c, t)
E(k, x) 2 {0, 1}128 //
encryptthenMAC ciphertext
Each of the length fields on line (⇤) is a 64bit value indicating the length in bytes of the respective field. If the input nonce N is not 96bits long, then N is padded to the closest multiple of 128 bits, yielding the padded string N 0 , and the initial counter value x is computed as x GHASH km , (N 0 k length(N )) which is a value in {0, 1}128 . As usual, the integrity tag t can be truncated to whatever length is desired. The shorter the tag t the more vulnerable the system becomes to ciphertext integrity attacks. Messages to be encrypted must be less than 232 blocks each (i.e., messages must be in X v for some v < 232 ). Recommendations in the standard suggest that a single key k should not be used to encrypt more than 232 messages. The GCM decryption algorithm takes as input a key k 2 K, a ciphertext (c, t), associate data d E(k, 0n ) and checks the and a nonce N . It operates as in encryptthenMAC: it first derives km CarterWegman integrity tag t. If valid it outputs the counter mode decryption of c. We emphasize that decryption must be atomic: no plaintext data is output before the integrity tag is verified over the entire message. GHASH. It remains to describe the keyed hash function GHASH defined over (X , X ` , X ). This hash function is used in a CarterWegman MAC and therefore, for security, must be a DUF. In Section 7.4 we showed that the function Hxpoly is a DUF and GHASH is essentially the same thing. Recall that Hxpoly (k, z) works by evaluating a polynomial derived from z at the point k. We described Hxpoly using arithmetic modulo a prime p so that both blocks of z and the output are elements in Zp . The hash function GHASH is almost the same as Hxpoly , except that the input message blocks and the output are elements of {0, 1}128 . Also, the DUF property holds with respect to the XOR operator , rather than subtraction modulo some number. As discussed in Remark 7.4, to build an XORDUF we use polynomials defined over the finite field GF(2128 ). This is a field of 2128 362
elements called a Galois field, which is where GCM gets its name. This field is defined by the irreducible polynomial g(X) := X 128 + X 7 + X 2 + X + 1. Elements of GF(2128 ) are polynomials over GF(2) of degree less than 128, with arithmetic done modulo g(X). While that sounds fancy, an element of GF(2128 ) can be conveniently represented as a string of 128 bits (each bit encodes one of the coefficients of the polynomial). Addition in the field is just XOR, while multiplication is a bit more complicated, but still not too difficult (see below — many modern computers provide direct hardware support). v With this notation, for k 2 GF(2128 ) and z 2 GF(2128 ) the function GHASH(k, z) is simply polynomial evaluation in GF(2128 ): GHASH(k, z) := z[0]k v + z[1]k v
1
+ . . . + z[v
1]k 2 GF(2128 )
(9.18)
That’s it. Appending the two length fields to the GHASH input on line (⇤) ensures that the XORDUF property is maintained even for messages of di↵erent lengths. Security. The AEAD security of GCM is similar to the analysis we did for generic composition of encryptthenMAC (Theorem 9.4), and follows from the security of the underlying block cipher as a PRF. The main di↵erence between GCM and our generic composition is that GCM “cuts a few corners” when it comes to keys: it uses just a single key k and uses E(k, 0n ) as the GHASH key, and E(k, x) as the pad that is used to mask the output of GHASH, which is similar to, but not exactly the sames as, what is done in CarterWegman. Importantly, the counter mode encryption begins with the counter value x0 := x + 1, so that the inputs to the PRF that are used to encrypt the message are guaranteed to be distinct from the inputs used to derive the GHASH key and pad. The above discussion focused on the case where the nonce is 96 bits. The other case, where GHASH is applied to the nonce to compute x, requires a more involved analysis — see Exercise 9.14. GCM has no nonce reuse resistance. If a nonce is accidentally reused on two di↵erent messages then all secrecy for those message is lost. Even worse, the GHASH secret key km is exposed (Exercise 7.13) and this can be used to break ciphertext integrity. Hence, it is vital that nonces not be reused in GCM. Optimizations and performance. There are many ways to optimize the implementation of GCM and GHASH. In practice, the polynomial in (9.18) is evaluated using Horner’s method so that processing each block of plaintext requires only one addition and one multiplication in GF(2128 ). Intel recently added a special instruction (called PCLMULQDQ) to their instruction set to quickly carry out binary polynomial multiplication. This instruction cannot be used directly to implement GHASH because of incompatibility with how the standard represents elements in GF(2128 ). Fortunately, work of Gueron shows how to overcome these difficulties and use the PCLMULQDQ instruction to speedup GHASH on Intel platforms. Since GHASH needs only one addition and one multiplication in GF(2128 ) per block one would expect that the bulk of the time during GCM encryption and decryption is spent on AES in counter mode. However, due to improvements in hardware implementations of AES, especially pipelining of the AESNI instructions, this is not always the case. On Intel’s Haswell processors (introduced in 2013) GCM is about three times slower than pure counter mode due to the extra overhead of GHASH. However, upcoming improvements in the implementation of PCLMULQDQ will likely make GCM just slightly more expensive than pure counter mode, which is the best one can hope for. 363
We should point out that it already is possible to implement secure authenticated encryption at a cost that is not much more than the cost of AES counter mode — this can be achieved using an integrated scheme such as OCB (see Exercise 9.17).
9.8
Case study: the TLS 1.3 record protocol
The Transport Layer Security (TLS) protocol is by far the most widely deployed security protocol. Virtually every online purchase is protected by TLS. Although TLS is primarily used to protect Web traffic, it is a general protocol that can protect many types of traffic: email, messaging, and many others. The original version of TLS was designed at Netscape where it was called the Secure Socket Layer protocol or SSL. SSL 2.0 was designed in 1994 to protect Web ecommerce traffic. SSL 3.0, designed in 1995, corrected several significant security problems in SSLv2. For example, SSL 2.0 uses the same key for both the cipher and the MAC. While this is bad practice — it invalidates the proofs of security for MtE and EtM — it also implies that if one uses a weak cipher key, say do to export restrictions, then the MAC key must also be weak. SSL 2.0 supported only a small number of algorithms and, in particular, only supported MD5based MACs. The Internet Engineering Task Force (IETF) created the Transport Layer Security (TLS) working group to standardize an SSLlike protocol. The working group produced a specification for the TLS 1.0 protocol in 1999 [31]. TLS 1.0 is a minor variation of SSL 3.0 and is often referred to as SSL version 3.1. TLS is supported by most major browsers and web servers and TLS 1.3 is the recommended protocol to use. We will mostly focus on TLS 1.3 here. The TLS 1.3 record protocol. Abstractly, TLS consists of two components. The first, called TLS session setup, negotiates the cipher suite that will be used to encrypt the session and then sets up a shared secret between the browser and server. The second, called the TLS record protocol uses this shared secret to securely transmit data between the two sides. TLS session setup uses publickey techniques and will be discussed later in Chapter 20. Here we focus on the TLS record protocol. In TLS terminology, the shared secret generated during session setup is called a mastersecret. This high entropy master secret is used to derive two keys kb!s and ks!b . The key kb!s encrypts messages from the browser to the server while ks!b encrypts messages in the reverse direction. TLS derives the two keys by using the master secret and other randomness as a seed for a key derivation function called HKDF (Section 8.10.5) to derive enough pseudorandom bits for the two keys. This step is carried out by both the browser and server so that both sides have the keys kb!s and ks!b . The TLS record protocol sends data in records whose size is at most 214 bytes. If one side needs to transmit more than 214 bytes, the record protocol fragments the data into multiple records each of size at most 214 . Each party maintains a 64bit write sequence number that is initialized to zero and is incremented by one for every record sent by that party. TLS 1.3 uses a noncebased AEAD cipher (E, D) to encrypt a record. Which noncebased AEAD cipher is used is determined by negotiation during TLS session setup. The AEAD encryption algorithm is given the following arguments: • secret key: kb!s or ks!b depending on whether the browser or server is encrypting. • plaintext data: up to 214 bytes. 364
• associated data: a concatenation of three fields: the encrypting party’s 64bit write sequence number, a 1byte record type (a value of 23 means application data), and a 2byte protocol version (set to 3.1 in TLS 3.1). • nonce (8 bytes or longer): the nonce is computed by (1) padding the encrypting party’s 64bit write sequence number on the left with zeroes to the expected nonce length and (2) XORing this padded sequence number with a random string (called client write iv or server write iv, depending on who is encrypting) that was derived from the master secret during session setup and is fixed for the life of the session. TLS 1.3 could have used an equivalent and slightly easier to comprehend method: choose the initial nonce value at random and then increment it sequentially for each record. The method used by TLS 1.3 is a little easier to implement. The AEAD cipher outputs a ciphertext c which is then formatted into an encrypted TLS record as follows: type
version
ciphertext c
length
where type is a 1byte record type (handshake record or application data record), version is a 2byte protocol version set to 3.1 for TLS 3.1, length is a 2byte field indicating the length of c, and c is the ciphertext. The type, version, and length fields are all sent in the clear. Notice that the nonce is not part of the encrypted TLS record. The recipient computes the nonce by itself. Why is the initial nonce value chosen at random? Why not simply set it to zero? In networking protocols the first message block sent over TLS is usually a fixed public value. If the nonce were set to zero then the first ciphertext would be computed as c0 E(k, m0 , d, 0) where the adversary knows m0 and associate data d. This opens up the system to an exhaustive search attack for the key k using a timespace tradeo↵ discussed in Chapter 18. The attack shows that with a large amount of precomputation and sufficient storage, an attacker can quickly recover k from c0 with nonnegligible advantage — for 128bit keys, such attacks may be feasible in the nottoodistant future. Randomizing the initial nonce “future proofs” TLS against such attacks. When a record is received, the receiving party runs the AEAD decryption algorithm to decrypt c. If decryption results in reject then the party sends a fatal bad record mac alert to its peer and shuts down the TLS session. The length field. In TLS 1.3, as in earlier versions of TLS, the record length is sent in the clear. Several attacks based on traffic analysis exploit record lengths to deduce information about the record contents. For example, if an encrypted TLS record contains one of two images of di↵erent size then the length will reveal to an eavesdropper which image was encrypted. Chen et al. [25] show that the lengths of encrypted records can reveal considerable information about private data that a user supplies to a cloud application. They use an online tax filing system as their example. Other works show attacks of this type on many other systems. Since there is no complete solution to this problem, it is often ignored. When encrypting a TLS record the length field is not part of the associated data and consequently has no integrity protection. The reason is that due to variable length padding, the length of c may not be known before the encryption algorithm terminates. Therefore, the length cannot be given as input to the encryption algorithm. This does not compromise security: a secure AEAD cipher will reject a ciphertext that is a result of tampering with the length field. 365
Replay prevention. An attacker may attempt to replay a previous record to cause the wrong action at the recipient. For example, the attacker could attempt to make the same purchase order be processed twice, by simply replaying the record containing the purchase order. TLS uses the 64bit sequence number to discard such replicated packets. TLS assumes inorder record delivery so that the recipient already knows what sequence number to expect without any additional information in the record. A replicated record will be discarded because the AEAD decryption algorithm will be given the wrong nonce as input.
9.9
Case study: an attack on nonatomic decryption in SSH
SSH (secure shell) is a popular command line tool for securely exchanging information with a remote host. SSH is designed to replace (insecure) UNIX tools such as telnet, rlogin, rsh, and rcp. Here we describe a fascinating vulnerability in an older cipher suite used in SSH. This vulnerability is an example of what can go wrong when decryption is not atomic, that is, when the decryption algorithm releases fragments of a decrypted record before verifying integrity of the entire record. First, a bit of history. The first version of SSH, called SSHv1, was made available in 1995. It was quickly pointed out that SSHv1 su↵ers from serious design flaws. • Most notably, SSHv1 provides data integrity by computing a Cyclic Redundancy Check (CRC) of the plaintext and appending the resulting checksum to the ciphertext in the clear. CRC is a simple keyless, linear function — so not only does this directly leak information about the plaintext, it is also not too hard to break integrity either. • Another issue is the incorrect use of CBC mode encryption. SSHv1 always sets the CBC initial value (IV) to 0. Consequently, an attacker can tell when two SSHv1 packets contain the same prefix. Recall that for CPA security one must choose the IV at random. • Yet another problem, the same encryption key was used for both directions (user to server and server to user). To correct these issues, a revised and incompatible protocol called SSHv2 was published in 1996. Session setup results in two keys ku!s , used to encrypt data from the user to the server, and ks!u , used to encrypt data in the reverse direction. Here we focus only how these keys are used for message transport in SSHv2. SSHv2 encryption. Let us examine an older cipher suite used in SSHv2. SSHv2 combines a CPAsecure cipher with a secure MAC using encryptandMAC (Exercise 9.10) in an attempt to construct a secure AEAD cipher. Specifically, SSHv2 encryption works as follows (Fig. 9.3): 1. Pad. Pad the plaintext with random bytes so that the total length of plaintext := packetlength k padlength k message k pad is a multiple of the cipher block length (16 bytes for AES). The pad length can be anywhere from 4 bytes to 255 bytes. The packet length field measures the length of the packet in bytes, not including the integrity tag or the packetlength field itself. 2. Encrypt. Encrypt the gray area in Fig. 9.3 using AES in randomized CBC mode with either ku!s or ks!u , depending on the encrypting party. SSHv2 uses a defective version of randomized CBC mode encryption described in Exercise 5.12. 366
Gray area is encrypted; Boxed area is authenticated by integrity tag packet len pad len message
pad
integrity tag
32 bits
Figure 9.3: An SSHv2 packet 3. MAC. A MAC is computed over a sequencenumber and the plaintext data in the thick box in Fig. 9.3. Here sequencenumber is a 32bit sequence number that is initialized to zero for the first packet, and is incremented by one after every packet. SSHv2 can use one of a number of MAC algorithms, but HMACSHA1160 must be supported. When an encrypted packet is received the decryption algorithm works as follows: first it decrypts the packetlength field using either ku!s or ks!u . Next, it reads that many more packets from the network plus as many additional bytes as needed for the integrity tag. Next it decrypts the rest of the ciphertext and verifies validity of the integrity tag. If valid, it removes the pad and returns the plaintext message. Although SSH uses encryptandMAC, which is not generally secure, we show in Exercise 9.10 that for certain combinations of cipher and MAC, including the required ones in SSHv2, encryptandMAC provides authenticated encryption. SSH boundary hiding via length encryption. An interesting aspect of SSHv2 is that the encryption algorithm encrypts the packet length field, as shown in Fig. 9.3. The motivation for this is to ensure that if a sequence of encrypted SSH packets are sent over an insecure network as a stream of bytes, then an eavesdropper should be unable to determine the number of packets sent or their lengths. This is intended to frustrate certain traffic analysis attacks that deduce information about the plaintext from its size. Hiding message boundaries between consecutive encrypted messages is outside the requirements addressed by authenticated encryption. In fact, many secure AEAD modes do not provide this level of secrecy. TLS 1.0, for example, sends the length of the every record in the clear making it easy to detect boundaries between consecutive encrypted records. Enhancing authenticated encryption 367
to ensure boundary hiding has been formalized by Boldyreva, Degabriele, Paterson, and Stam [20], proposing a number of constructions satisfying the definitions. An attack on nonatomic decryption. Notice that CBC decryption is done in two steps: first the 32bit packetlength field is decrypted and used to decide how many more bytes to read from the network. Next, the rest of the CBC ciphertext is decrypted. Generally speaking, AEAD ciphers are not designed to be used this way: plaintext data should not be used until the entire ciphertext decryption process is finished; however, in SSHv2 the decrypted length field is used before its integrity has been verified. Can this be used to attack SSHv2? A beautiful attack [1] shows how this nonatomic decryption can completely compromise secrecy. Here we only describe the highlevel idea, ignoring many details. Suppose an attacker intercepts a 16byte ciphertext block c and it wants to learn the first four bytes of the decryption of c. It does so by abusing the decryption process as follows: first, it sends the ciphertext block c to the server as if it were the first block of a new encrypted packet. The server decrypts c and interprets the first four bytes as a length field `. The server now expects to read ` bytes of data from the network before checking the integrity tag. The attacker can slowly send to the server arbitrary bytes, one byte at a time, waiting after each byte to see if the server responds. Once the server reads ` bytes it attempts to verify the integrity tag on the bytes it received and this most likely fails causing the server to send back an error message. Thus, once ` bytes are read the attacker receives an error message. This tells the attacker the value of ` which is what it wanted. In practice, there are many complications in mounting an attack like this. Nevertheless, it shows the danger of using decrypted data — the length field in this case — before its integrity has been verified. As mentioned above, we refer to [20] for encryption methods that securely hide packet lengths. A clever traffic analysis attack on SSH. SSHv2 operates by sending one network packet for every user keystroke. This gives rise to an interesting traffic analysis attack reported in [98]. Suppose a network eavesdropper knows that the user is entering a password at his or her keyboard. By measuring timing di↵erences between consecutive packets, the eavesdropper obtains timing information between consecutive keystrokes. This exposes information about the user’s password: a large timing gap between consecutive keystrokes reveals information about the keyboard position of the relevant keys. The authors show that this information can significantly speed up an o✏ine password dictionary attack. To make matters worse, password packets are easily identified since applications typically turn o↵ echo during password entry so that password packets do not generate an echo packet from the server. Some SSH implementations defend against this problem by injecting randomly timed “dummy” messages to make traffic analysis more difficult. Dummy messages are identified by setting the first message byte to SSH MSG IGNORE and are ignored by the receiver. The eavesdropper cannot distinguish dummy records from real ones thanks to encryption.
9.10
Case study: 802.11b WEP, a badly broken system
The IEEE 802.11b standard ratified in 1999 defines a protocol for short range wireless communication (WiFi). Security is provided by a Wired Equivalent Privacy (WEP) encapsulation of 802.11b 368
cleartext payload m L
CRC(m)
RC4( IV k k )
encrypted frame
IV
Figure 9.4: WEP Encryption data frames. The design goal of WEP is to provide data privacy at the level of a wired network. WEP, however, completely fails on this front and gives us an excellent case study illustrating how a weak design can lead to disastrous results. When WEP is enabled, all members of the wireless network share a long term secret key k. The standard supports either 40bit keys or 128bit keys. The 40bit version complies with US export restrictions that were in e↵ect at the time the standard was drafted. We will use the following notation to describe WEP: • WEP encryption uses the RC4 stream cipher. We let RC4(s) denote the pseudo random sequence generated by RC4 given the seed s. • We let CRC(m) denote the 32bit CRC checksum of a message m 2 {0, 1}⇤ . The details of CRC are irrelevant for our discussion and it suffices to view CRC as some fixed function from bit strings to {0, 1}32 . Let m be an 802.11b cleartext frame. The first few bits of m encode the length of m. To encrypt an 802.11b frame m the sender picks a 24bit IV and computes: c m k CRC(m) cfull (IV, c)
RC4(IV k k)
The WEP encryption process is shown in Fig. 9.4. The receiver decrypts by first computing c RC4(IV k k) to obtain a pair (m, s). The receiver accepts the frame if s = CRC(m) and rejects it otherwise. Attack 1: IV collisions. The designers of WEP understood that a stream cipher key should never be reused. Consequently, they used the 24bit IV to derive a perframe key kf := IV k k. The standard, however, does not specify how to choose the IVs and many implementations do so poorly. We say that an IV collision occurs whenever a wireless station happens to send two frames, say frame number i and frame number j, encrypted using the same IV. Since IVs are sent in the clear, an eavesdropper can easily detect IV collisions. Moreover, once an IV collision occurs the attacker can use the twotime pad attack discussed in Section 3.3.1 to decrypt both frames i and j. So, how likely is an IV collision? By the birthday paradox, an implementation that chooses p a random IV for each frame will cause an IV collision after only an expected 224 = 212 = 4096 frames. Since each frame body is at most 1156 bytes, a collision will occur after transmitting about 4MB on average. 369
Alternatively, an implementation could generate the IV using a counter. The implementation will exhaust the entire IV space after 224 frames are sent, which will take about a day for a wireless access point working at full capacity. Even worse, several wireless cards that use the counter method reset the counter to 0 during powerup. As a result, these cards will frequently reuse low value IVs, making the traffic highly vulnerable to a twotime pad attack. Attack 2: related keys. A far more devastating attack on WEP encryption results from the use of related RC4 keys. In Chapter 3 we explained that a new and random stream cipher key must be chosen for every encrypted message. WEP, however, uses keys 1 k k, 2 k k, . . . which are all closely related — they all have the same suffix k. RC4 was never designed for such use, and indeed, is completely insecure in these settings. Fluhrer, Mantin, and Shamir [38] showed that after about a million WEP frames are sent, an eavesdropper can recover the entire long term secret key k. The attack was implemented by Stubblefield, Ioannidis, and Rubin [101] and is now available in a variety of hacking tools such as WepCrack and AirSnort. Generating per frame keys should have been done using a PRF, for example, setting the key for frame i to ki := F (k, IV) — the resulting keys would be indistinguishable from random, independent keys. Of course, while this approach would have prevented the related keys problem, it would not solve the IV collision problem discussed above, or the malleability problem discussed next. Attack 3: malleability. Recall that WEP attempts to provide authenticated encryption by using a CRC checksum for integrity. In a sense, WEP uses the MACthenencrypt method, but it uses CRC instead of a MAC. We show that despite the encryption step, this construction utterly fails to provide ciphertext integrity. The attack uses the linearity of CRC. That is, given CRC(m) for some message m, it is easy to compute CRC(m ) for any . More precisely, there is a public function L such that for any m ` and 2 {0, 1} we have that CRC(m
) = CRC(m)
L( )
This property enables an attacker to make arbitrary modifications to a WEP ciphertext without ever being detected by the receiver. Let c be a WEP ciphertext, namely c = m, CRC(m) For any
RC4(IV k k)
2 {0, 1}` , an attacker can create a new ciphertext c0 c0 = RC4(IV k k) RC4(IV k k)
RC4(IV k k)
c
, L( ) , which satisfies
m, CRC(m)
, L( ) =
m
, CRC(m)
L( ) =
m
, CRC(m
)
. We see that given the encryption of m, an attacker Hence, c0 decrypts without errors to m can create a valid encryption of m for any of his choice. We explained in Section 3.3.2 that this can lead to serious attacks. Attack 4: Chosen ciphertext attack. The protocol is vulnerable to a chosen ciphertext attack called chopchop that lets the attacker decrypt an encrypted frame of its choice. We describe a simple version of this attack in Exercise 9.5. 370
IPsec
Internet
IPsec
gateway
1
2
gateway
3
4
west branch
5
6
east branch
Figure 9.5: A virtual private network (VPN) between east and west office branches Attack 5: Denial of Service. We briefly mention that 802.11b su↵ers from a number of serious Denial of Service (DoS) attacks. For example, in 802.11b a wireless client sends a “disassociate” message to the wireless station once the client is done using the network. This allows the station to free memory resources allocates to that client. Unfortunately, the “disassociate” message is unauthenticated, allowing anyone to send a disassociate message on behalf of someone else. Once disassociated, the victim will take a few seconds to reestablish the connection to the base station. As a result, by sending a single “disassociate” message every few seconds, an attacker can prevent a computer of their choice from connecting to the wireless network. These attacks are implemented in 802.11b tools such as Void11. 802.11i. Following the failures of the 802.11b WEP protocol, a new standard called 802.11i was ratified in 2004. 802.11i provides authenticated encryption using a MACthenencrypt mode called CCM. In particular, CCM uses (raw) CBCMAC for the MAC and counter mode for encryption. Both are implemented in 802.11i using AES as the underlying PRF. CCM was adopted by NIST as a federal standard [86].
9.11
Case study: IPsec
The IPsec protocol provides confidentiality and integrity for Internet IP packets. The protocol was first published in 1998 and was subsequently updated in 2005. The IPsec protocol consists of many subprotocols that are not relevant for our discussion here. In this section we will focus on the most commonly used IPsec protocol called encapsulated security payload (ESP) in tunnel mode. Virtual private networks (VPNs) are an important application for IPsec. A VPN enables two office branches to communicate securely over a public Internet channel, as shown in Fig. 9.5. Here, packets from machines 1,2,3 are encrypted at the west gateway using IPsec and transmitted over the public channel. The east gateway decrypts each received packet and forwards it to its destination inside the east branch, namely, one of 4,5,6. We note that all packets sent from west to east are encrypted using the same cryptographic key kw!e . Packets sent from east to west are processed similarly, but encrypted using a di↵erent key, ke!w . We will use this VPN example as our motivating example for IPsec. To understand IPsec one first needs a basic understanding of the IP protocol. Here we focus on IP version 4 (IPv4), which is currently widely deployed. The left side of Fig. 9.6 shows a (cleartext) 371
IPsec ESP packet Gray area is encrypted Boxed area is authenticated by integrity tag
ver
packet len
prot=ESP source IP address
cleartext IP packet ver
dest IP address
packet len
security parameters index (SPI) sequence number protocol
hdr checksum
source IP address
packet
dest IP address
padding
payload
pad len integrity tag
32 bits 32 bits
Figure 9.6: Cleartext IPv4 packet and an IPsec ESP packet
372
next hdr
IPv4 packet. The packet consists of a packet header and a packet payload. The header contains a bunch of fields, but only a few are relevant to our discussion: • The first four bits indicate the version number which is set to 4 for IPv4. • The 2byte packet length field contains the length in bytes of the entire packet including the header. • The 1byte protocol field describes the packet payload For example, protocol = 6 indicates a TCP payload. • the 2byte header checksum contains a checksum of all header bytes (excluding the checksum field). The checksum is used to detect random transmission errors in the header. Packets with an invalid checksum are dropped at the recipient. The checksum can be computed by anyone and consequently provides no integrity against an attacker. In fact, Internet routers regularly change fields in the packet header as the packet moves from router to router and recompute the checksum. • The source and destination IP indicate the source and destination addresses for the packet. • The payload contains the packet contents and is variable length. IPsec encapsulated security payload (ESP). The right side of Fig. 9.6 shows the result of encrypting a packet with ESP in tunnel mode. We first describe the fields in the encrypted packet and then describe the encryption process. IPsec key management — the SPI field. Every ESP endpoint maintains a security association database (SAD). A record in the SAD is called a security association (SA) and is identified by a 32 bit identifier called a security parameters index (SPI). A SAD record (an SA) contains many connectionspecific parameters, such as the ESP encryption algorithm (e.g. 3DESCBC or AESCBC), the ESP secret key (e.g. kw!e or ke!w ), the source and destination IP addresses, the SPI, and various keyexchange parameters. When the east branch gateway sends out a packet, it uses the packet’s destination IP address and other parameters to choose a security association (SA) in its security association database (SAD). The gateway embeds the 32bit SPI of the chosen SA in the packet header and encrypts the packet using the secret key specified in the SA. When the packet arrives at its destination, the recipient locates an appropriate SA in its own SAD using the following algorithm: 1. First, look for an SA matching the received (SPI, dest address, source address); 2. If no match is found, the recipient looks for a match based on the (SPI, dest address) pair; 3. Otherwise, it looks for a match based on the SPI only. If no SA exists for the received packet, the packet is discarded. Otherwise, the gateway decrypts the packet using the secret key specified in the chosen SA. Most often an SA is used for transmitting packets in one direction, e.g., from east to west. A bidirectional TCP connection between east and west uses two separate SAs — one for packets from east to west and one for packets from west to east. Generally, an ESP endpoint maintains two SAD records for each peer. The SAD at a particular host is managed semimanually. Some parameters are managed manually while others are negotiated between the communicating hosts. In particular, an SA secret 373
key can be set manually at both endpoints or it can be negotiated using an IPsec key exchange protocol called IKE [62]. We will not discuss SAD management here. ESP antireplay — the sequence number field. The sequence number enables the recipient to detect and discard duplicate packets. Duplication can result from a network error or can be caused by an attacker who is deliberately replaying old packets. Every ESP end point maintains a sequence number for each security association. By default the sequence number is 64 bits long (called an extended sequence number), although older versions of ESP use a shorter 32 bit sequence number. The sequence number is initialized to zero when the security association is created and is incremented by one for each packet sent using the SA. The entire 64 bits are included in the MAC calculation. However, only the 32 least significant bits (LSB) are included in the ESP packet header. In other words, ESP endpoints maintain 64bit counters, of which the 32 MSBs are implicit while the 32 LSBs are explicit in the packet header. For our discussion of sequence numbers, we assume that there is at most a single host sending packets for each security association (SA). Hence, for a particular SA there is no danger of two hosts sending a packet with the same sequence number. Note that multiple hosts can receive packets for a particular SA, as in the case of multicast. We only disallow multiple hosts from sending packets using a single SA. For a particular SA, the recipient must discard any packet that contains a 32bit sequence number that was previously contained in an earlier packet. Since packets can arrive out of order, verifying sequence number unicity at the recipient takes some e↵ort. RFC 4303 recommends that the recipient maintain a window (e.g. bit vector) of size 32. The “right” edge of the window represents the highest, validated sequence number value received on this SA. Packets that contain sequence numbers lower than the “left” edge of the window are discarded. Received packets falling within the window are checked against the list of received packets within the window, and are discarded if their sequence number was already seen. The window shifts whenever a valid packet with a sequence number on the “right” of the current window is received. Consequently, the receiver recovers gracefully from a long sequence of lost packets If more than 232 consecutive packets are lost, then the 64bit sequence numbers at the sender and receiver will go out of sync — the 32 MSBs implicitly maintained by the two will di↵er. As a result, all further packets will be rejected due to MAC validation failure. This explains why the designers of ESP chose to include 32 bits in the packet header — a loss of 232 packets in unlikely. Including fewer bits (e.g. 16 bits) would have greatly increased the chance of communication failure. Padding and the next header field. ESP first appends a pad to ensure that the length of the data to encrypt is a multiple of the block length of the chosen encryption algorithm (e.g. a multiple of 16 bytes for AESCBC). It also ensures that the resulting ciphertext length is a multiple of four bytes. The pad length is anywhere from 0 to 255 bytes. An additional padlength byte is appended to indicate the number of padding bytes preceding it. Finally, a next header (nexthdr) byte, is appended to indicate the payload type. Most often the payload type is an IPv4 packet in which case nexthdr=4. ESP supports an optional traffic flow confidentiality (TFC) service where the sender attempts to hide the length of the plaintext packet. To do so, the sender appends dummy (unspecified) bytes to the payload before padding takes place. The length of the TFC pad is arbitrary. The packet length field in the plaintext IP header indicates the beginning of the TFC pad. The TFC pad is removed after decryption. ESP also supports “dummy” packets to defeat traffic analysis. The goal is to prevent an observer 374
from telling when the sender transmits data. For example, one can instruct the sender to transmit a packet every millisecond, whether it has data to send or not. When no data is available, the sender transmits a “dummy” packet which is indicated by setting nexthdr=59. Since the nexthdr field is encrypted an observer cannot tell dummy packets from real packets. However, at the destination, all dummy packets are discarded immediately after decryption. The encryption process. discuss each step in turn.
ESP implements the encryptthenMAC method in four steps. We
1. Pad. The pad, including the optional TFC pad and next header field, are appended to the plaintext IP packet. 2. Encrypt. The gray area in Fig. 9.6 is encrypted with the algorithm and key specified by the SA. ESP supports a variety of encryption algorithms, but is required to support 3DESCBC, AESCBC, and AES counter mode. For CBC modes the IV is prepended to the encrypted payload and is sent in the clear. The encryption algorithm can be set to NULL in which case no encryption takes place. This is used when ESP provides integrity but no confidentiality. 3. MAC. An integrity tag is computed using an algorithm and key specified in the SA. The tag is computed over the following data SPI k 64bit sequence number k ciphertext where ciphertext is the result of Step 2. Note that the tag is computed over the 64 bit sequence number even though only 32 bits are embedded in the packet. The resulting tag is placed in the integrity tag field following the ciphertext. ESP supports a variety of MAC algorithms, but is required to support HMACSHA196, HMACMD596, and AESXCBCMAC96 (XCBCMAC is a variant of CMAC). The integrity tag field is optional and is omitted if the encryption algorithm already provides authenticated encryption, as in the case of GCM. 4. Encapsulate. Finally, an IPv4 packet header is prepended to obtain an ESP packet as shown on the right side of Fig. 9.6. The protocol field in the IPv4 header is set to 50 indicating an ESP payload. Decryption follows a similar process. The recipient first checks the 32bit sequence number. If the value is repeated or outside the allowed window, the packet is dropped. Next, the recipient checks the tag field, and rejects the packet if MAC verification fails. The packet is then decrypted and the padding removed. If the packet is a dummy packet (i.e. the next header field is equal to 59), the packet is discarded. Finally, the original cleartext packet is reconstructed and sent to the destination. Note that in principle, the sequence number field could have been encrypted. The designers of ESP chose to send the field in the clear so as to reduce the time until a duplicate packet is rejected. Security. IP packets can arrive at any order, be duplicated, and even modified. By relying on encryptthenMAC and on the sequence number, ESP ensures that the recipient sees a data stream identical to the one transmitted by the sender. One issue that haunts ESP is a setting that provides CPAsecure encryption without an integrity check. RFC 4303 states that 375
ESP allows encryptiononly SAs because this may o↵er considerably better performance and still provide adequate security, e.g., when higherlayer authentication/integrity protection is o↵ered independently. Relying on a higher application layer for integrity is highly risky. On the sender side the application layer processes data before passing it to the IP layer. Hence, this implements MACthenencrypt which from a theoretical point view we know can be insecure. More importantly, in practice it is dangerous to assume that the higher layer will protect the entire IP packet. For example, a higher layer such as SSL may provide integrity without encryption. Combining encryptiononly ESP and integrityonly SSL will be insecure since the SSL layer will not provide integrity for the encrypted packet header. As a result, an attacker can tamper with the destination IP field in the encrypted packet. The recipient’s IPsec gateway will decrypt the packet and forward the result to an unintended destination, thus causing a serious privacy breach. This and other dangers of the ESP encryptiononly mode are discussed in [8, 87]. We note, however, that when the cipher used provides authenticated encryption (such as GCM mode) it is perfectly fine to use encryption without an integrity check, since the cipher already provides authenticated encryption.
9.12
A fun application: private information retrieval
To be written.
9.13
Notes
Citations to the literature to be added.
9.14
Exercises
9.1 (AEsecurity: simple examples). Let (E, D) be an AEsecure cipher. Consider the following derived ciphers: ( D(k, c1 ) if D(k, c1 ) = D(k, c2 ) (a) E1 (k, m) := E(k, m), E(k, m) ; D2 k, (c1 , c2 ) := reject otherwise ( D(k, c1 ) if c1 = c2 (b) E2 (k, m) := c E(k, m), output (c, c) ; D2 k, (c1 , c2 ) := reject otherwise Show that part (b) is AEsecure, but part (a) is not. 9.2 (AEsecurity: some insecure constructions). Let (E, D) be a CPAsecure cipher defined over (K, M, C) and let H1 : M ! T and H2 : C ! T be collision resistant hash functions. Define
376
the following two ciphers: E1 (k, m) := E(k, m), H1 (m) ;
E2 (k, m) := E(k, m), H2 (c) ;
(
D(k, c1 ) reject ( D(k, c1 ) D2 k, (c1 , c2 ) := reject
D1 k, (c1 , c2 ) :=
if H1 (D(k, c1 )) = c2 otherwise if H2 (c1 ) = c2 otherwise
Show that both ciphers are not AEsecure. 9.3 (An Android Keystore Attack). Let (E, D) be a secure block cipher defined over (K, X ) and let (Ecbc , Dcbc ) be the cipher derived from (E, D) using randomized CBC mode, as in Section 5.4.3. Let H : M ! X be a collision resistant hash function. Consider the following attempt at building an AEsecure cipher: ⇢ (t, m) Dcbc (k, c) D1 (k, c) := E1 (k, m) := Ecbc k, (H(m), m) ; if t = H(m) output m, otherwise reject Show that (E1 , D1 ) is not AEsecure by giving a chosenciphertext attack on it. You may assume m 2 X for simplicity. This construction was used to protect secret keys in the Android KeyStore. The chosenciphertext attack resulted in a compromise of the key store [93]. 9.4 (Redundant message encoding does not give AE). The attack in the previous exercise can be generalized if instead of using CBC encryption as the underlying cipher, we use randomized counter mode, as in Section 5.4.2. Let (Ectr , Dctr ) be such a countermode cipher, and assume 0 0 that its message space is {0, 1}` . Let f : {0, 1}` ! {0, 1}` be a onetoone function, and let 0 g : {0, 1}` ! {0, 1}` [ {?} be its inverse, in the sense that g(m0 ) = m whenever m0 = f (m) for some m, and g(m0 ) = ? if m0 is not in the image of f . Intuitively, f represents an “error detecting ˜ 0 , this code”: a message m 2 {0, 1}` is “encoded” as m0 = f (m). If m0 gets modified into a value m 0 modification will be detected if g(m ˜ ) = ?. Now define a new cipher (E2 , D2 ) with message space ` {0, 1} as follows: ⇢ 0 m Dctr (k, c) E2 (k, m) := Ectr k, f (m) ; D1 (k, c) := if g(m0 ) 6= ? output g(m0 ), otherwise reject Show that (E2 , D2 ) is not AEsecure by giving a chosenciphertext attack on it. 9.5 (Chopchop attack). The parity bit b for a message m 2 {0, 1}⇤ is just the XOR of all the bits in m. After appending the parity bit, the message m0 = m k b has the property that the XOR of all the bits is zero. Parity bits are sometimes used as a very simple form of error detection. They are meant to provide a little protection against lowprobability, random errors: if a single bit of m0 gets flipped, this can be detected, since the XOR of the bits of the corrupted m0 will now be one. Consider a cipher where encryption is done using randomized counter mode without any padding. Messages are variable length bit strings and ciphertexts are bit strings of the same length as plaintext. No MAC is used, but before the plaintext is encrypted, the sender appends a parity bit to the end of the plaintext. After the receiver decrypts, he checks the parity bit and returns either the plaintext (with the parity bit removed) or reject. Design a chosenciphertext attack that recovers the complete plaintext of every encrypted message. 377
Hint: Use the fact that the system encrypts variable length messages. Remark: A variant of this attack, called chopchop, was used successfully against encryption in the 802.11b protocol. The name is a hint for how the attack works. Note that the previous exercise already tells us that this scheme is not CCAsecure, but the attack in this exercise is much more devastating. 9.6 (Nested encryption). Let (E, D) be an AEsecure cipher. Consider the following derived cipher (E 0 , D0 ): ( D k1 , D(k2 , c) if D(k2 , c) 6= reject 0 0 D (k1 , k2 ), c := E (k1 , k2 ), m := E k2 , E(k1 , m) ; reject otherwise (a) Show that (E 0 , D0 ) is AEsecure even if the adversary knows k1 , but not k2 . (b) Show that (E 0 , D0 ) is not AEsecure if the adversary knows k2 but not k1 . (c) Design a cipher built from (E, D) where keys are pairs (k1 , k2 ) 2 K2 and the cipher remains AEsecure even if the adversary knows one of the keys, but not the other. 9.7 (A format oracle attack). Let E be an arbitrary CPAsecure cipher, and assume that the key space for E is {0, 1}n . Show how to “sabotage” E to obtain another cipher E 0 such that E 0 is still CPA secure, but E 0 is insecure against chosen ciphertext attack, in the following sense. In the attack, the adversary is allowed to make several decryption queries, such that in each query, the adversary only learns whether the result of the decryption was reject or not. Design an adversary that makes a series of decryption queries as above, and then outputs the secret key in its entirety. . 9.8 (Choose independent keys). Let us see an example of a CPAsecure cipher and a secure MAC that are insecure when used in encryptthenMAC when the same secret key k is used for both the cipher and the MAC. Let (E, D) be a block cipher defined over (K, X ) where X = {0, 1}n and X  is superpoly. Consider randomized CBC mode encryption built from (E, D) as the CPAsecure cipher for single block messages: an encryption of m 2 X is the pair c := (r, E(k, r m)) where r is the random IV. Use RawCBC built from (E, D) as the secure MAC. This MAC is secure in this context because it is only being applied fixed length messages (messages in X 2 ): the tag on a ciphertext c 2 X 2 is t := E k, E(k, c[0]) c[1] . Show that using the same key k for both the cipher and the MAC in encryptthenMAC results in a cipher that is not CPA secure. 9.9 (MACthenencrypt). Prove that MACthenencrypt provides authenticated encryption when the underlying cipher is randomized CBC mode encryption and the MAC is a secure MAC. For concreteness, if the underlying cipher works on blocks of a fixed size, a message m is a sequence of full blocks, and the tag t for the MAC is one full block, so the message that is CBCencrypted is the block sequence m k t. 9.10 (An AEAD from encryptandMAC). Let (E, D) be randomized counter mode encryption defined over (K, M, C) where the underlying secure PRF has domain X . We let E(k, m; r) denote the encryption of message m with key k using r 2 X as the IV. Let F be a secure PRF defined over (K, (M⇥D ⇥ N ), X ). Show that the following cipher (E1 , D1 ) is a secure noncebased 378
AEAD cipher assuming X  is superpoly. E1 (ke , km ), m, d, D1 (ke , km ), (c, t), d,
N N)
:= t F km , (m, d, N ) , c R E(kc , m; t), output (c, t) ⇢ m D(ke , c; t) := if F km , (m, d, N ) 6= t output reject, otherwise output m
This method is loosely called encryptandMAC because the message m is both encrypted by the cipher and is the input to the MAC signing algorithm, which here is a PRF. Discussion: This construction is related to the authenticated SIV cipher (Exercise 9.11) and o↵ers similar nonce reuse resistance. One downside of this system is that the tag t cannot be truncated as one often does with a PRFbased MAC. 9.11 (Authenticated SIV). We discuss a modification of the SIV construction, introduced in Exercise 5.8, that provides ciphertext integrity without enlarging the ciphertext any further. We call this the authenticated SIV construction. With E = (E, D), F , and E 0 = (E 0 , D0 ) as in Exercise 5.8, we define E 00 = (E 0 , D00 ), where ⇢ m D(k, c) 00 0 := D (k, k ), c if E 0 ((k, k 0 ), m) = c output m, otherwise output reject Assume that R is superpoly and that for very fixed key k 2 K and m 2 M, the function E(k, m; ·) : R ! C is one to one (which holds for counter and CBC mode encryption). Show that E 00 provides ciphertext integrity.
Note: Since the encryption algorithm of E 00 is the same as that of E 0 we know that E 00 is deterministic CPAsecure, assuming that E is CPAsecure (as was shown in Exercise 5.8). 9.12 (Constructions based on strongly secure block ciphers). Let (E, D) be a block cipher defined over (K, M ⇥ R). (a) As in Exercise 5.6, let (E 0 , D0 ) be defined as E 0 (k, m) := r
R
R
R, c
D0 (k, c) := (m, r0 )
E k, (m, r) , output c
D(k, c), output m
Show that (E 0 , D0 ) is CCAsecure provided (E, D) is a strongly secure block cipher and 1/R is negligible. This is an example of a CCAsecure cipher that clearly does not provide ciphertext integrity. (b) Let (E 00 , D00 ) be defined as E 00 (k, m) := r R R, c R E k, (m, r) , output (c, r) ⇢ (m, r0 ) D(k, c) 00 D k, (c, r) := if r = r0 output m, otherwise output reject This cipher is defined over K, M, (M⇥R)⇥R . Show that (E 00 , D00 ) is AEsecure provided (E, D) is a strongly secure block cipher and 1/R is negligible. 379
(c) Suppose that 0 2 R and we modify algorithms E 00 and D00 to work as follows: ˜ 00 (k, m) := r 0, c R E k, (m, r) , output c E ⇢ (m, r0 ) D(k, c) 00 ˜ D k, c := if r0 = 0 output m, otherwise output reject ˜ 00 ) is onetime AEsecure provided (E, D) is a strongly secure block cipher, ˜ 00 , D Show that (E and 1/R is negligible. 9.13 (MAC from encryption). Let (E, D) be a cipher defined over (K, M, C). Define the following MAC system (S, V ) also defined over (K, M, C): ( accept if D(k, t) = m S(k, m) := E(k, m); V (k, m, t) := reject otherwise Show that if (E, D) has ciphertext integrity then (S, V ) is a secure MAC system. 9.14 (GCM analysis). Give a complete security analysis of GCM (see Section 9.7). Show that it is noncebased AEAD secure assuming the security of the underlying block cipher as a PRF and that GHASH is an XORDUF. Start out with the easy case when the nonce is 96bits. Then proceed to the more general case where GHASH may be applied to the nonce to compute x. 9.15 (Plaintext integrity). Consider a weaker notion of integrity called plaintext integrity, or simply PI. The PI game is identical to the CI game except that the winning condition is relaxed to: • D(k, c) 6= reject, and • D(k, c) 62 {m1 , m2 , . . .} Prove that the following holds: (a) Show that MACthenEncrypt is both CPA and PI secure. Note: The MACthenEncrypt counterexample (Section 9.4.2) shows that a system that is CPA and PI secure is not CCAsecure (and, therefore, not AEsecure). (b) Prove that a system that is CCA and PIsecure is also AEsecure. The proof only needs a weak version of CCA, namely where the adversary issues a single decryption query and is told whether the ciphertext is accepted or rejected. Also, you may assume a superpolysized message space. 9.16 (Encrypted UHF MAC). Let H be a hash function defined over (KH , M, X ) and (E, D) be a cipher defined over (KE , X , C). Define the encrypted UHF MAC system I = (S, V ) as follows: for key (k1 , k2 ) and message m 2 M define S (k1 , k2 ), m := E k1 , H(k2 , m) ( accept if H(k2 , m) = D(k1 , c), V (k1 , k2 ), m, c := reject otherwise.
380
Show that I is a secure MAC system assuming H is a computational UHF and (E, D) provides authenticated encryption. Recall from Section 7.4 that CPA security of (E, D) is insufficient for this MAC system to be secure. 9.17 (Simplified OCB mode). OCB is an elegant and efficient AE cipher built from a tweakable block cipher (as defined in Exercise 4.11). Let (E, D) be a tweakable block cipher defined over (K, X , T ) where X := {0, 1}n and the tweak set is T := N ⇥ { `, . . . , `}. Consider the following noncebased cipher (E 0 , D0 ) with key space K, message space X ` , ciphertext space X `+1 , and nonce space N . For simplicity, the cipher does not support associated data. E 0 (k, m, N ) := 8 > create (uninitialized) c 2 X m > > > > checksum 0n > > > > > < for i = 0, . . . , m 1 : c[i] E k, m[i], (N , i + 1) > > > checksum checksum m[i] > > > > > t E k, checksum, (N , m) > > : output (c, t)
D0 (k, (c, t), N ) := 8 > create (uninitialized) m 2 X c > > > > checksum 0n > > > > > < for i = 0, . . . , c 1 : m[i] D k, c[i], (N , i + 1) > > > checksum checksum m[i] > > > > 0 > E k, checksum, (N , c) t > > : if t = t0 output m, else reject
9 > > > > > > > > > > = > > > > > > > > > > ;
9 > > > > > > > > > > = > > > > > > > > > > ;
(a) Prove that (E 0 , D0 ) is a noncebased AEsecure cipher assuming (E, D) is a strongly secure tweakable block cipher and X  is superpoly. (b) Show that if t were computed as t E k, checksum, (N , 0) then the scheme would be insecure: it would have no ciphertext integrity. 9.18 (Noncommitting encryption). Let (E, D) be a cipher. We say that the cipher is noncommitting if an adversary can find a ciphertext c and two keys k0 , k1 such that c decrypts successfully under both k0 and k1 and the resulting plaintexts are di↵erent. The noncommitting property means that the adversary can transmit c, but if he or she are later required to reveal the decryption key, say for an internal audit, the adversary can “open” the ciphertext in two di↵erent ways. (a) Let (E, D) be an encryptthenMAC AEsecure cipher where the underlying encryption is randomized counter mode built using a secure PRF. Show that (E, D) is noncommitting. (b) Show that GCM mode encryption is noncommitting. (c) Describe a simple way in which the ciphers from parts (a) and (b) can be made committing. 9.19 (Middlebox encryption). In this exercise we develop a mode of encryption that lets a middlebox placed between the sender and recipient inspect all traffic in the clear, but prevents the middlebox for modifying traffic enroute. This is often needed in enterprise settings where a middlebox ensures that no sensitive information is accidentally sent out. Towards this goal let us define a middlebox cipher as a tuple of four algorithms (E, D, D0 , K) where E(k, m) and D(k, c) are the usual encryption and decryption algorithms used by the endpoints, K is an algorithm that derives a subkey k 0 from the primary key k (i.e., k 0 R K(k)), and D0 (k 0 , c) is the decryption algorithm used by the middlebox with the subkey k 0 . We require the usual correctness properties: D(k, c) and D0 (k 0 , c) output m whenever c R E(k, m) and k 0 R K(k). 381
(a) Security for a middlebox cipher (E, D, D0 , K) captures our desired confidentiality and integrity requirements. In particular, we say that a middlebox cipher is secure if the following three properties hold: (i) the cipher is secure against a chosen plaintext attack (CPA security) when the adversary knows nothing about k, (ii) the cipher provides ciphertext integrity with respect to the decryption algorithm D 0 (k 0 , ·), and the adversary knows nothing about k, and (iii) the cipher provides ciphertext integrity with respect to the decryption algorithm D(k, ·), and the adversary is given a subkey k 0 R K(k), but again knows nothing about k. The second requirement says that the middlebox will only decrypt authentic ciphertexts. The third requirement says that the receiving endpoint will only decrypt authentic ciphertexts, even if the middlebox is corrupt. Formalize these requirements as attack games. (b) Give a construction that satisfies your definition from part (a). You can use an AE secure cipher and a secure MAC as building blocks.
382
Part II
Public key cryptography
383
In the second part of the book we study how parties who don’t share a secret key can communicate over a public network. We start o↵ by introducing the basic tools used in public key cryptography — the RSA and DiffieHellman functions. We then show how one party, Alice, can send messages to another party, Bob, given Bob’s public key. We then discuss digital signatures and given several constructions. Some constructions are based entirely on tools tools from Part I while other constructions are based on public key tools. The last two chapters in part II explain how to establish a secure session using identification and key exchange.
384
Chapter 10
Public key tools We begin our discussion of publickey cryptography by introducing several basic tools that will be used in the remainder of the book. The main applications for these tools will emerge in the next few chapters where we use them for publickey encryption, digital signatures, and key exchange. Since we use some basic algebra and number theory in this chapter, the reader is advised to first briefly scan through Appendix A. We start with a simple toy problem: generating a shared secret key between two parties so that a passive eavesdropping adversary cannot feasibly guess their shared key. The adversary can listen in on network traffic, but cannot modify messages enroute or inject his own messages. In a later chapter we develop the full machinery needed for key exchange in the presence of an active attacker who may tamper with network traffic. At the onset we emphasize that security against eavesdropping is typically not sufficient for real worldapplications, since an attacker capable of listening to network traffic is often also able to tamper with it; nevertheless, this toy eavesdropping model is a good way to introduce the new publickey tools.
10.1
A toy problem: anonymous key exchange
Two users, Alice and Bob, who never met before talk on the phone. They are worried that an eavesdropper is listening to their conversation and hence they wish to encrypt the session. Since Alice and Bob never met before they have no shared secret key with which to encrypt the session. Thus, their initial goal is to generate a shared secret unknown to the adversary. They may later use this secret as a sessionkey for secure communication. To do so, Alice and Bob execute a protocol where they take turns in sending messages to each other. The eavesdropping adversary can hear all these messages, but cannot change them or inject his own messages. At the end of the protocol Alice and Bob should have a secret that is unknown to the adversary. The protocol itself provides no assurance to Alice that she is really talking to Bob, and no assurance to Bob that he is talking to Alice — in this sense, the protocol is “anonymous.” More precisely, we model Alice and Bob as communicating machines. A key exchange protocol P is a pair of probabilistic machines (A, B) that take turns in sending messages to each other. At the end of the protocol, when both machines terminate, they both obtain the same value k. A protocol transcript TP is the sequence of messages exchanged between the parties in one execution of the protocol. Since A and B are probabilistic machines, we obtain a di↵erent transcript 385
every time we run the protocol. Formally, the transcript TP of protocol P is a random variable, which is a function of the random bits generated by A and B. The eavesdropping adversary A sees the entire transcript TP and its goal is to figure out the secret k. We define security of a key exchange protocol using the following game. Attack Game 10.1 (Anonymous key exchange). For a key exchange protocol P = (A, B) and a given adversary A, the attack game runs as follows. • The challenger runs the protocol between A and B to generate a shared key k and transcript TP . It gives TP to A. • A outputs a guess kˆ for k. We define A’s advantage, denoted AnonKEadv[A, P ], as the probability that kˆ = k. 2 Definition 10.1. We say that an anonymous key exchange protocol P is secure against an eavesdropper if for all efficient adversaries A, the quantity AnonKEadv[A, P ] is negligible. This definition of security is extremely weak, for three reasons. First, we assume the adversary is unable to tamper with messages. Second, we only guarantee that the adversary cannot guess k in its entirety. This does not rule out the possibility that the adversary can guess, say, half the bits of k. If we are to use k as a secret session key, the property we would really like is that k is indistinguishable from a truly random key. Third, the protocol provides no assurance of the identities of the participants. We will strengthen Definition 10.1 to meet these stronger requirements in Chapter 20. Given all the tools we developed in Part 1, it is natural to ask if anonymous key exchange can be done using an arbitrary secure symmetric cipher. The answer is yes, it can be done as we show in Section 10.8, but the resulting protocol is highly inefficient. To develop efficient protocols we must first introduce a few new tools.
10.2
Oneway trapdoor functions
In this section, we introduce a tool that will allow us to build an efficient and secure key exchange protocol. In Section 8.11, we introduced the notion of a oneway function. This is a function F : X ! Y that is easy to compute, but hard to invert. As we saw in Section 8.11, there are a number of very efficient functions that are plausibly oneway. Oneway functions, however, are not sufficient for our purposes. We need oneway functions with a special feature, called a trapdoor. A trapdoor is a secret that allows one to efficiently invert the function; however, without knowledge of the trapdoor, the function remains hard to invert. Let us make this notion more precise. Definition 10.2 (Trapdoor function scheme). Let X and Y be finite sets. A trapdoor function scheme T , defined over (X , Y), is a triple of algorithms (G, F, I), where • G is a probabilistic key generation algorithm that is invoked as (pk , sk ) called a public key and sk is called a secret key.
R
G(), where pk is
• F is a deterministic algorithm that is invoked as y F (pk , x), where pk is a public key (as output by G) and x lies in X . The output y is an element of Y. 386
• I is a deterministic algorithm that is invoked as x I(sk , y), where sk is a secret key (as output by G) and y lies in Y. The output x is an element of X . Moreover, the following correctness property should be satisfied: for all possible outputs (pk , sk ) of G(), and for all x 2 X , we have I(sk , F (pk , x) ) = x. Observe that for every pk , the function F (pk , ·) is a function from X to Y. The correctness property says that sk is the trapdoor for inverting this function; note that this property also implies that the function F (pk , ·) is onetoone. Note that we do not insist that F (pk , ·) maps X onto Y. That is, there may be elements y 2 Y that do not have any preimage under F (pk , ·). For such y, we make no requirements on algorithm I — it can return some arbitrary element x 2 X (one might consider returning a special reject symbol in this case, but it simplifies things a bit not to do this). In the special case where X = Y, then F (pk , ·) is not only onetoone, but onto. That is, F (pk , ·) is a permutation on the set X . In this case, we may refer to (G, F, I) as a trapdoor permutation scheme defined over X . The basic security property we want from a trapdoor permutation scheme is a onewayness property, which basically says that given pk and F (pk , x) for random x 2 X , it is hard to compute x without knowledge of the trapdoor sk . This is formalized in the following game. Attack Game 10.2 (Oneway trapdoor function scheme). For a given trapdoor function scheme T = (G, F, I), defined over (X , Y), and a given adversary A, the attack game runs as follows: • The challenger computes (pk , sk )
R
G(),
x
R
X,
y
F (pk , x)
and sends (pk , y) to the adversary. • The adversary outputs x ˆ 2 X. We define the adversary’s advantage in inverting T , denoted OWadv[A, T ], to be the probability that x ˆ = x. 2 Definition 10.3. We say that a trapdoor function scheme T is one way if for all efficient adversaries A, the quantity OWadv[A, T ] is negligible. Note that in Attack Game 10.2, since the value x is uniformly distributed over X and F (pk , ·) is onetoone, it follows that the value y := F (pk , x) is uniformly distributed over the image of F (pk , ·). In the case of a trapdoor permutation scheme, where X = Y, the value of y is uniformly distributed over X .
10.2.1
Key exchange using a oneway trapdoor function scheme
We now show how to use a oneway trapdoor function scheme T = (G, F, I), defined over (X , Y), to build a secure anonymous key exchange protocol. The protocol runs as follows, as shown in Fig. 10.1: • Alice computes (pk , sk )
R
G(), and sends pk to Bob.
• Upon receiving pk from Alice, Bob computes x 387
R
X,y
F (pk , x), and sends y to Alice.
Alice (pk , sk )
R
Bob G()
x
pk y
x
R
R
X
F (pk , x)
x
I(pk , y)
Figure 10.1: Key exchange using a trapdoor function scheme • Upon receiving y from Bob, Alice computes x
I(sk , y).
The correctness property of the trapdoor function scheme guarantees that at the end of the protocol, Alice and Bob have the same value x — this is their shared, secret key. Now consider the security of this protocol, in the sense of Definition 10.1. In Attack Game 10.1, the adversary sees the transcript consisting of the two messages pk and y. If the adversary could compute the secret x from this transcript with some advantage, then this very same adversary could be used directly to break the trapdoor function scheme, as in Attack Game 10.2, with exactly the same advantage.
10.2.2
Mathematical details
We give a more mathematically precise definition of a trapdoor function scheme, using the terminology defined in Section 2.4. Definition 10.4 (Trapdoor function scheme). A trapdoor function scheme is a triple of efficient algorithms (G, F, I) along with families of spaces with system parameterization P : X = {X As usual, that
2Z
1
,⇤ } ,⇤ , Y
= {Y
,⇤ } ,⇤ .
is a security parameter and ⇤ 2 Supp(P ( )) is a domain parameter. We require
1. X is efficiently recognizable and sampleable. 2. Y is efficiently recognizable. 3. G is an efficient probabilistic algorithm that on input , ⇤, where 2 Z 1 , ⇤ 2 Supp(P ( )), outputs a pair (pk , sk ), where pk and sk are bit strings whose lengths are always bounded by a polynomial in . 4. F is an efficient deterministic algorithm that on input , ⇤, pk , x, where 2 Z 1, ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)) for some sk , and x 2 X ,⇤ , outputs an element of Y ,⇤ .
388
5. I is an efficient deterministic algorithm that on input , ⇤, sk , y, where 2 Z 1, ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)) for some pk , and y 2 Y ,⇤ , outputs an element of X ,⇤ . 6. For all 2 Z 1 , ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)), and x 2 X I( , ⇤; sk , F ( , ⇤; pk , x)) = x.
,⇤ ,
we have
As usual, in defining the onewayness security property, we parameterize Attack Game 10.2 by the security parameter , and the advantage OWadv[A, T ] is actually a function of . Definition 10.3 should be read as saying that OWadv[A, T ]( ) is a negligible function.
10.3
A trapdoor permutation scheme based on RSA
We now describe a trapdoor permutation scheme that is plausibly oneway. It is called RSA after its inventors, Rivest, Shamir, and Adleman. Recall that a trapdoor permutation is a special case of a trapdoor function, where the domain and range are the same set. This means that for every publickey, the function is a permutation of its domain, which is why we call it a trapdoor permutation. Despite many years of study, RSA is essentially the only known reasonable candidate trapdoor permutation scheme (there are a few others, but they are all very closely related to the RSA scheme). Here is how RSA works. First, we describe a probabilistic algorithm RSAGen that takes as input an integer ` > 2, and an odd integer e > 2. RSAGen(`, e) := generate a random `bit prime p such that gcd(e, p generate a random `bit prime q such that gcd(e, q n pq d e 1 mod (p 1)(q 1) output (n, d).
1) = 1 1) = 1 and q 6= p
To efficiently implement the above algorithm, we need an efficient algorithm to generate random `bit primes. This is discussed in ??. Also, we use the extended Euclidean algorithm (see ??) to compute e 1 mod (p 1)(q 1). Note that since gcd(e, p 1) = gcd(e, q 1) = 1, it follows that gcd(e, (p 1)(q 1)) = 1, and hence e has a multiplicative inverse modulo (p 1)(q 1). Now we describe the RSA trapdoor permutation scheme TRSA = (G, F, I). It is parameterized by fixed values of ` and e. • Key generation runs as follows: G() :=
(n, d) R RSAGen(`, e), output (pk , sk ).
pk
(n, e),
sk
(n, d)
• For a given public key pk = (n, e), and x 2 Zn , we define F (pk , x) := xe 2 Zn . • For a given secret key sk = (n, d), and y 2 Zn , we define I(sk , y) := y d 2 Zn . Note that although the encryption exponent e is considered to be a fixed system parameter, we also include it as part of the public key pk .
389
A technicality. For each fixed pk = (n, e), the function F (pk , ·) maps Zn into Zn ; thus, the domain and range of this function actually vary with pk . However, in our definition of a trapdoor permutation scheme, the domain and range of the function are not allowed to vary with the public key. So in fact, this scheme does not quite satisfy the formal syntactic requirements of a trapdoor permutation scheme. One could easily generalize the definition of a trapdoor permutation scheme, to allow for this. However, we shall not do this; rather, we shall state and analyze various schemes based on a trapdoor permutation scheme as we have defined it, and then show how to instantiate these schemes using RSA. Exercise 10.23 explores an idea that builds a proper trapdoor permutation scheme based on RSA. Ignoring this technical issue for the moment, let us first verify that TRSA satisfies the correctness requirement of a trapdoor permutation scheme. This is implied by the following: Theorem 10.1. Let n = pq where p and q are distinct primes. Let e and d be integers such that ed ⌘ 1 (mod (p 1)(q 1)). Then for all x 2 Z, we have xed ⌘ x (mod n). Proof. The hypothesis that ed ⌘ 1 (mod (p 1)(q 1)) just means that ed = 1 + k(p 1)(q 1) for some integer k. Certainly, if x ⌘ 0 (mod p), then xed ⌘ 0 ⌘ x (mod p); otherwise, if x 6⌘ 0 (mod p), then by Fermat’s little theorem (see ??), we have xp
1
⌘1
and so xed ⌘ x1+k(p
1)(q 1)
⌘ x · x(p
(mod p),
1) k(q 1)
⌘ x · 1k(q
1)
⌘x
(mod p).
Therefore, xed ⌘ x
(mod p).
xed ⌘ x
(mod q).
By a symmetric argument, we have
Thus, xed x is divisible by the distinct primes p and q, and must therefore be divisible by their product n, which means xed ⌘ x (mod n). 2 So now we know that TRSA satisfies the correctness property of a trapdoor permutation scheme. However, it is not clear that it is oneway. For TRSA , onewayness means that there is no efficient algorithm that given n and xe , where x 2 Zn is chosen at random, can e↵ectively compute x. It is clear that if TRSA is oneway, then it must be hard to factor n; indeed, if it were easy to factor n, then one could compute d in exactly the same way as is done in algorithm RSAGen, and then use d to compute x = y d . It is widely believed that factoring n is hard, provided ` is sufficiently large — typically, ` is chosen to be between 1000 and 1500. Moreover, the only known efficient algorithm to invert TRSA is to first factor n and then compute d as above. However, there is no known proof that the assumption that factoring n is hard implies that TRSA is oneway. Nevertheless, based on current evidence, it seems reasonable to conjecture that TRSA is indeed oneway. We state this conjecture now as an explicit assumption. As usual, this is done using an attack game. Attack Game 10.3 (RSA). For given integers ` > 2 and odd e > 2, and a given adversary A, the attack game runs as follows: 390
• The challenger computes (n, d)
R
RSAGen(`, e),
R
x
Zn ,
y
xe 2 Z n
and gives the input (n, y) to the adversary. • The adversary outputs x ˆ 2 Zn . We define the adversary’s advantage in breaking RSA, denoted RSAadv[A, `, e], as the probability that x ˆ = x. 2 Definition 10.5 (RSA assumption). We say that the RSA assumption holds for (`, e) if for all efficient adversaries A, the quantity RSAadv[A, `, e] is negligible. We analyze the RSA assumption and present several known attacks on it later on in Chapter 15. We next introduce some terminology that will be useful later. Suppose (n, d) is an output of RSAGen(`, e), and suppose that x 2 Zn and let y := xe . The number n is called an RSA modulus, the number e is called an encryption exponent, and the number d is called a decryption exponent. We call (n, y) an instance of the RSA problem, and we call x a solution to this instance of the RSA problem. The RSA assumption asserts that there is no efficient algorithm that can e↵ectively solve the RSA problem.
10.3.1
Key exchange based on the RSA assumption
Consider now what happens when we instantiate the key exchange protocol in Section 10.2.1 with TRSA . The protocol runs as follows: • Alice computes (n, d)
R
RSAGen(`, e), and sends (n, e) to Bob.
• Upon receiving (n, e) from Alice, Bob computes x • Upon receiving y from Bob, Alice computes x
R
Zn , y
xe , and sends y to Alice.
yd.
The secret shared by Alice and Bob is x. The message flow is the same as in Fig. 10.1. Under the RSA assumption, this is a secure anonymous key exchange protocol.
10.3.2
Mathematical details
We give a more mathematically precise definition of the RSA assumption, using the terminology defined in Section 2.4. In Attack Game 10.3, the parameters ` and e are actually polybounded and efficiently computable functions of a security parameter . Likewise, RSAadv[A, `, e] is a function of . As usual, Definition 10.5 should be read as saying that RSAadv[A, `, e]( ) is a negligible function. There are a couple of further wrinkles we should point out. First, as already mentioned above, the RSA scheme does not quite fit our definition of a trapdoor permutation scheme, as the definition of the latter does not allow the set X to vary with the public key. It would not be too difficult to modify our definition of a trapdoor permutation scheme to accommodate this generalization. Second, the specification of RSAGen requires that we generate random prime numbers of a given bit length. In theory, it is possible to do this in (expected) polynomial time; however, the most practical algorithms (see Section ??) may — with negligible probability — output a number that is 391
not a prime. If that should happen, then it may be the case that the basic correctness requirement — namely, that I(sk , F (pk , x)) = x for all pk , sk , x — is no longer satisfied. It would also not be too difficult to modify our definition of a trapdoor permutation scheme to accommodate this type of generalization as well. For example, we could recast this requirement as an attack game (in which any efficient adversary wins with negligible probability): in this game, the challenger generates (pk , sk ) R G() and sends (pk , sk ) to the adversary; the adversary wins the game if he can output x 2 X such that I(sk , F (pk , x)) 6= x. While this would be a perfectly reasonable definition, using it would require us to modify security definitions for higherlevel constructs. For example, if we used this relaxed correctness requirement in the context of key exchange, we would have to allow for the possibility that the two parties end up with di↵erent keys with some negligible probability.
10.4
DiffieHellman key exchange
In this section, we explore another approach to constructing secure key exchange protocols, which was invented by Diffie and Hellman. Just as with the protocol based on RSA, this protocol will require a bit of algebra and number theory. However, before getting in to the details, we provide a bit of motivation and intuition. Consider the following “generic” key exchange protocol the makes use of two functions E and F . Alice chooses a random secret ↵, computes E(↵), and sends E(↵) to Bob over an insecure channel. Likewise, Bob chooses a random secret , computes E( ), and sends E( ) to Alice over an insecure channel. Alice and Bob both somehow compute a shared key F (↵, ). In this highlevel description, E and F are some functions that should satisfy the following properties: 1. E should be easy to compute; 2. given ↵ and E( ), it should be easy to compute F (↵, ); 3. given E(↵) and , it should be easy to compute F (↵, ); 4. given E(↵) and E( ), it should be hard to compute F (↵, ). Properties 1–3 ensure that Alice and Bob can efficiently implement the protocol: Alice computes the shared key F (↵, ) using the algorithm from Property 2 and her given data ↵ and E( ). Bob computes the same key F (↵, ) using the algorithm from Property 3 and his given data E(↵) and . Property 4 ensures that the protocol is secure: an eavesdropper who sees E(↵) and E( ) should not be able to compute the shared key F (↵, ). Note that properties 1–4 together imply that E is hard to invert; indeed, if we could compute efficiently ↵ from E(↵), then by Property 2, we could efficiently compute F (↵, ) from E(↵), E( ), which would contradict Property 4. To make this generic approach work, we have to come up with appropriate functions E and F . To a first approximation, the basic idea is to implement E in terms of exponentiation to some fixed base g, defining E(↵) := g ↵ and F (↵, ) := g ↵ . Notice then that E(↵) = (g ↵ ) = F (↵, ) = (g )↵ = E( )↵ . Hence, provided exponentiation is efficient, Properties 1–3 are satisfied. Moreover, if Property 4 is to be satisfied, then at the very least, we require that taking logarithms (i.e., inverting E) is hard. 392
To turn this into a practical and plausibly secure scheme, we cannot simply perform exponentiation on ordinary integers since the numbers would become too large. Instead, we have to work in an appropriate finite algebraic domain, which we introduce next.
10.4.1
The key exchange protocol
Suppose p is a large prime and that q is a large prime dividing p 1 (think of p as being very large random prime, say 2048 bits long, and think of q as being about 256 bits long). We will be doing arithmetic mod p, that is, working in Zp . Recall that Z⇤p is the set of nonzero elements of Zp . An essential fact is that since q divides p 1, Z⇤p has an element g of order q (see Section ??). This means that g q = 1 and that all of the powers g a , for a = 0, . . . , q 1, are distinct. Let G := {g a : a = 0, . . . , q 1}, so that G is a subset of Z⇤p of cardinality q. It is not hard to see that G is closed under multiplication and inversion; that is, for all u, v 2 G, we have uv 2 G and u 1 2 G. Indeed, g a · g b = g a+b = g c with c := (a + b) mod q, and (g a ) 1 = g d with d := ( a) mod q. In the language of algebra, G is called a subgroup of the group Z⇤p . For every u 2 G and integers a and b, it is easy to see that ua = ub if a ⌘ b mod q. Thus, the value of ua depends only on the residue class of a modulo q. Therefore, if ↵ = [a]q 2 Zq is the residue class of a modulo q, we can define u↵ := ua and this definition is unambiguous. From here on we will frequently use elements of Zq as exponents applied to elements of G. So now we have everything we need to describe the DiffieHellman key exchange protocol. We assume that the description of G, including g 2 G and q, is a system parameter that is generated once and for all at system setup time and shared by all parties involved. The protocol runs as follows, as shown in Fig. 10.2: R
1. Alice computes ↵ 2. Bob computes
R
Zq , u Zq , v
g ↵ , and sends u to Bob. g and sends v to Alice.
3. Upon receiving v from Bob, Alice computes w
v↵
4. Upon receiving u from Alice, Bob computes w
u
The secret shared by Alice and Bob is w = v↵ = g↵ = u .
10.4.2
Security of DiffieHellman key exchange
For a fixed element g 2 G, di↵erent from 1, the function from Zq to G that sends ↵ 2 Zq to g ↵ 2 G is called the discrete exponentiation function. This function is onetoone and onto, and its inverse function is called the discrete logarithm function, and is usually denoted Dlogg ; thus, for u 2 G, Dlogg (u) is the unique ↵ 2 Zq such that u = g ↵ . The value g is called the base of the discrete logarithm. If the DiffieHellman protocol has any hope of being secure, it must be hard to compute ↵ from ↵ g for a random ↵; in other words, it must be hard to compute the discrete logarithm function. There are a number of candidate group families G where the discrete logarithm function is believed to be hard to compute. For example, when p and q are sufficiently large, suitably chosen primes, 393
G, g, q
G, g, q
Alice
Bob
↵
R
Zq
R
u
g↵ v
w
v ↵ = g xy
Zq
g
w
u = g xy
Figure 10.2: DiffieHellman key exchange the discrete logarithm function in the order q subgroup of Z⇤p is believed to be hard to compute (p should be at least 2048bits, and q should be at least 256bits). This assumption is called the discrete logarithm assumption and is defined in the next section. Unfortunately, the discrete logarithm assumption by itself is not enough to ensure that the DiffieHellman protocol is secure. Observe that the protocol is secure if and only if the following holds: given g ↵ , g 2 G, where ↵
R
Zq and
R
Zq , it is hard to compute g ↵ 2 G.
This security property is called the computational DiffieHellman assumption. Although the computational DiffieHellman assumption is stronger than the discrete logarithm assumption, all evidence still suggests that this is a reasonable assumption in groups where the discrete logarithm assumption holds.
10.5
Discrete logarithm and related assumptions
In this section, we state the discrete logarithm and related assumptions more precisely and in somewhat more generality, and explore in greater detail relationships among them. The subset G of Z⇤p that we defined above in Section 10.4 is a specific instance of a general type of mathematical object known as a cyclic group. There are in fact other cyclic groups that are very useful in cryptography, most notably, groups based on elliptic curves — we shall study elliptic curve cryptography in Chapter 16. From now on, we shall state assumptions and algorithms in terms of an abstract cyclic group G of prime order q generated by g 2 G. In general, such groups may be selected by a randomized process, and again, the description of G, including g 2 G and q, is a system parameter that is generated once and for all at system setup time and shared by all parties involved. We shall use just a bit of terminology from group theory. The reader who is unfamiliar with the concept of a group may wish to refer to ??; alternatively, for the time being, the reader may simply ignore this abstraction entirely: • Whenever we refer to a “cyclic group,” the reader may safely assume that this means the specific set G defined above as a subgroup of Z⇤p . 394
• The “order of G” is just a fancy name for the size of the set G, which is q. • A “generator of G” is an element g 2 G with the property that every element of G can be expressed as a power of g. We begin with a formal statement of the discrete logarithm assumption, stated in our more general language. As usual, we need an attack game. Attack Game 10.4 (Discrete logarithm). Let G be a cyclic group of prime order q generated by g 2 G. For a given adversary A, define the following attack game: • The challenger computes
↵
R
Zq ,
u
g↵,
and gives the value u to the adversary. • The adversary outputs some ↵ ˆ 2 Zq .
We define A’s advantage in solving the discrete logarithm problem for G, denoted DLadv[A, G], as the probability that ↵ ˆ = ↵. 2 Definition 10.6 (Discrete logarithm assumption). We say that the discrete logarithm (DL) assumption holds for G if for all efficient adversaries A the quantity DLadv[A, G] is negligible. We say that g ↵ is an instance of the discrete logarithm (DL) problem (for G), and that ↵ is a solution to this problem instance. By convention, we assume that the description of G includes its order q and a generator g. The DL assumption asserts that there is no efficient algorithm that can e↵ectively solve the DL problem. Note that the DL assumption is defined in terms of a group G and generator g 2 G. As already mentioned, the group G and generator g are chosen and fixed at system setup time via a process that may be randomized. Also note that all elements of G \ {1} are in fact generators for G, but we do not insist that g is chosen uniformly among these (but see Exercise 10.16). Di↵erent methods for selecting groups and generators give rise to di↵erent DL assumptions (and the same applies to the CDH and DDH assumptions, defined below). Now we state the computational DiffieHellman assumption. Attack Game 10.5 (Computational DiffieHellman). Let G be a cyclic group of prime order q generated by g 2 G. For a given adversary A, the attack game runs as follows. • The challenger computes ↵,
R
Zq ,
u
g↵,
v
g ,
w
g↵
and gives the pair (u, v) to the adversary. • The adversary outputs some w ˆ 2 G.
We define A’s advantage in solving the computational DiffieHellman problem for G, denoted CDHadv[A, G], as the probability that w ˆ = w. 2 Definition 10.7 (Computational DiffieHellman assumption). We say that the computational DiffieHellman (CDH) assumption holds for G if for all efficient adversaries A the quantity CDHadv[A, G] is negligible. 395
We say that (g ↵ , g ) is an instance of the computational DiffieHellman (CDH) problem, and that g ↵ is a solution to this problem instance. Again, by convention, we assume that the description of G includes its order q and a generator g. The CDH assumption asserts that there is no efficient algorithm that can e↵ectively solve the CDH problem. An interesting property of the CDH problem is that there is no general and efficient algorithm to even recognize correct solutions to the CDH problem, that is, given an instance (u, v) of the CDH problem, and a group element w, ˆ to determine if w ˆ is a solution to the given problem instance. This is in contrast to the RSA problem: given an instance (n, e, y) of the RSA problem, and an element x ˆ of Z⇤n , we can efficiently test if x ˆ is a solution to the given problem instance simply by testing if x ˆe = y. In certain cryptographic applications, this lack of an efficient algorithm to recognize solutions to the CDH problem can lead to technical difficulties. However, this apparent limitation is also an opportunity: if we assume not only that solving the CDH problem is hard, but also that recognizing solutions to CDH problem is hard, then we can sometimes prove stronger security properties for certain cryptographic schemes. We shall now formalize the assumption that recognizing solutions to the CDH problem is hard. In fact, we shall state a stronger assumption, namely, that even distinguishing solutions from random group elements is hard. It turns out that this stronger assumption is equivalent to the weaker one (see Exercise 10.9). Attack Game 10.6 (Decisional DiffieHellman). Let G be a cyclic group of prime order q generated by g 2 G. For a given adversary A, we define two experiments. Experiment b
(b = 0, 1):
• The challenger computes ↵, ,
R
Zq ,
u
g↵,
v
g ,
w0
g↵ ,
w1
g ,
and gives the triple (u, v, wb ) to the adversary. • The adversary outputs a bit ˆb 2 {0, 1}. If Wb is the event that A outputs 1 in Experiment b, we define A’s advantage in solving the decisional DiffieHellman problem for G as DDHadv[A, G] := Pr[W0 ]
Pr[W1 ] .
2
Definition 10.8 (Decisional DiffieHellman assumption). We say that the decisional DiffieHellman (DDH) assumption holds for G if for all efficient adversaries A the quantity DDHadv[A, G] is negligible. For ↵, , 2 Zq , we call (g ↵ , g , g ) a DHtriple if = ↵ ; otherwise, we call it a nonDHtriple. The DDH assumption says that there is no efficient algorithm that can e↵ectively distinguish between random DHtriples and random triples. More precisely, in the language of Section 3.11, the DDH assumptions says that the uniform distribution over DHtriples and the uniform distribution over G3 are computationally indistinguishable. It is not hard to show the the DDH assumption implies that it is hard to distinguish between random DHtriples and random nonDHtriples (see Exercise 10.6). 396
Clearly, the DDH assumption implies the CDH assumption: if we could e↵ectively solve the CDH problem, then we could easily determine if a given triple (u, v, w) ˆ is a DHtriple by first computing a correct solution w to the instance (u, v) of the CDH problem, and then testing if w = w. ˆ In defining the DL, CDH, and DDH assumptions, we have restricted our attention to prime order groups. This is convenient for a number of technical reasons. See, for example, Exercise 10.20, where you are asked to show that the DDH assumption for groups of even order is simply false.
10.5.1
Random selfreducibility
An important property of the discretelog function in a group G is that it is either hard almost everywhere in G or easy everywhere in G. A middle ground where discretelog is easy for some inputs and hard for others is not possible. We prove this by showing that the discretelog function has a random self reduction. Consider a specific cyclic group G of prime order q generated by g 2 G. Suppose A is an efficient algorithm with the following property: if u 2 G is chosen at random, then Pr[A(u) = Dlogg (u)] = ✏. That is, on a random input u, algorithm A computes the discrete logarithm of u with probability ✏. Here, the probability is over the random choice of u, as well as any random choices made by A itself.1 Suppose ✏ = 0.1. Then the group G is of little use in cryptography since an eavesdropper can use A to break 10% of all DiffieHellman key exchanges. However, this does not mean that A is able to compute Dlogg (u) with nonzero probability for all u 2 G. It could be the case that for 10% of the inputs u 2 G, algorithm A always computes Dlogg (u), while for the remaining 90%, it never computes Dlogg (u). We show how to convert A into an efficient algorithm B with the following property: for all u 2 G, algorithm B on input u successfully computes Dlogg (u) with probability ✏. Here, the probability is only over the random choices made by B. We so do using a reduction that maps a given discretelog instance to a random discretelog instance. Such a reduction is called a random self reduction. Theorem 10.2. Consider a specific cyclic group G of prime order q generated by g 2 G. Suppose A is an efficient algorithm with the following property: if u 2 G is chosen at random, then Pr[A(u) = Dlogg (u)] = ✏, with the probability is over the random choice of u and the random choices made by A. Then there is an efficient algorithm B with the following property: for all u 2 G, algorithm B either outputs fail or Dlogg (u), and it outputs the latter with probability ✏, where now the probability is only over the random choices made by B. Theorem 10.2 implements the transformation shown in Fig. 10.3. The point is that, unlike A, algorithm B works for all inputs. To compute discretelog of a particular u 2 G one can iterate B on the same input u several times, say nd1/✏e times for some n. Using the handy inequality 1 + x exp(x) (which holds for all x), this iteration will produce the discretelog with probability 1 (1 ✏)nd1/✏e 1 exp( n). In particular, if 1/✏ is polybounded, we can efficiently compute the discrete logarithm of any group element with negligible failure probability. In contrast, iterating A on the same input u many times may never produce a correct answer. Consequently, if discretelog is easy for a nonnegligible fraction of instances, then it will be easy for all instances. 1
Technical note: the probability ✏ is not quite the same as DLadv[A, G], as the latter is also with respect to the random choice of group/generator made at system setup time; here, we are viewing these as truly fixed.
397
G
G
=)
A works for inputs here
B works everywhere
Figure 10.3: The e↵ect of a random self reduction Proof of Theorem 10.2. Algorithm B works as follows: Input: u 2 G Output: Dlogg (u) or fail R
u1 ↵1
Zq u·g 2G A(u1 )
if g ↵1 = 6 u1 then output fail else output ↵ ↵1 Suppose that u = g ↵ . Observe that u1 = g ↵+ . Since is uniformly distributed over Zq , the group element u1 is uniformly distributed over G. Therefore, on input u1 , adversary A will output = ↵, and otherwise, B ↵1 = ↵ + with probability ✏. When this happens, B will output ↵1 will output fail. 2 Why random self reducibility is important. Any hard problem can potentially form the basis of a cryptosystem. For example, an NPhard problem known as subset sum has attracted attention for many years. Unfortunately, many hard problems, including subset sum, are only hard in the worst case. Generally speaking, such problems are of little use in cryptography, where we need problems that are not just hard in the worst case, but hard on average (i.e., for randomly chosen inputs). For a problem with a random selfreduction, if it hard in the worst case, then it must be hard on average. This implication makes such problems attractive for cryptography. One can also give random self reductions for both the CDH and DDH problems, as well as for the RSA problem (in a more limited sense). These ideas are developed the chapter exercises.
10.5.2
Mathematical details
As in previous sections, we give the mathematical details pertaining to the DL, CDH, and DDH assumptions. We use the terminology introduced in Section 2.4. This section may be safely skipped on first reading with very little loss in understanding. To state the assumptions asymptotically we introduce a security parameter that identifies the group in which the DL, CDH, and DDH games are played. We will require that the adversary’s advantage in breaking the assumption is a negligible function of . As lambda increases the adversary’s advantage in breaking discretelog in the group defined by should quickly go to zero. 398
To make sense of the security parameter we need a family of groups that increase in size as increases. As in Section 2.4, this family of groups is parameterized by both and an additional system parameter ⇤. The idea is that once is chosen, a system parameter ⇤ is generated by a system parameterization algorithm P . The pair ( , ⇤) then fully identifies the group G ,⇤ where the DL, CDH, and DDH games are played. Occasionally we will refer to ⇤ as a group description. This ⇤ is a triple ⇤ := ( ⇤1 , q, g ) where ⇤1 is an arbitrary string, q is prime number that represents the order of the group G g is a generator of G ,⇤ .
,⇤ ,
and
Definition 10.9 (group family). A group family G consists of an algorithm Mul along with a family of spaces: G = {G ,⇤ } ,⇤ with system parameterization algorithm P , such that 1. G is efficiently recognizable. 2. Algorithm Mul is an efficient deterministic algorithm that on input u, v 2 G ,⇤ , outputs w 2 G ,⇤ .
2Z
1,
⇤ 2 Supp(P ( )),
3. For all 2 Z 1 , ⇤ = (⇤1 , q, g) 2 Supp(P ( )), algorithm Mul is a multiplication operation on G ,⇤ that defines a cyclic group of prime order q generated by g. The definition implies that all the spaces G ,⇤ are efficiently sampleable. Since ⇤ = (⇤1 , q, g) we can randomly sample a random element u of G ,⇤ by picking a random ↵ R Zq and setting u g ↵ . Specific group families may allow for a more efficient method that generates a random group element. The group identity element may always be obtained by raising g to the power q, although for specific group families, there are most likely simpler and faster ways to do this. An example. We define the asymptotic version of a subgroup of prime order q within Z⇤p , where q is a prime dividing p 1, and p itself is prime. Here the system parameterization algorithm P takes as input and outputs a group description ⇤ := (p, q, g) where p is a random `( )bit prime (for some polybounded length function `) and g is an element of Z⇤p of order q. The group G ,⇤ is the subgroup of Z⇤p generated by g. Elements of G ,⇤ may be efficiently recognized as follows: first, one can check that a given bit string properly encodes an element u of Z⇤p ; second, one can check that uq = 1. Armed with the concept of a group family, we now parameterize the DL Attack Game 10.4 by the security parameter . In that game, the adversary is given the security parameter and a group description ⇤ = (⇤1 , q, g), where g is a generator for the group G ,⇤ . It is also given a random u 2 G ,⇤ , and it wins the game if it computes Dlogg (u). Its advantage DLadv[A, G] is now a function of , and for each , this advantage is a probability that depends on the random choice of group and generator, as well as the random choices made by the the challenger and adversary. Definition 10.6 should be read as saying that DLadv[A, G]( ) is a negligible function. We use the same approach to define the asymptotic CDH and DDH assumptions.
399
10.6
Collision resistant hash functions from numbertheoretic primitives
It turns out that the RSA and DL assumptions are extremely versatile, and can be used in many cryptographic applications. As an example, in this section, we show how to build collisionresistant hash functions based on the RSA an DL assumptions. Recall from Section 8.1 that a hash function H defined over (M, T ) is an efficiently computable function from M to T . In most applications, we want the message space M to be much larger than the digest space T . We also defined a notion of collision resistance, which says that for every efficient adversary A, its collisionfinding advantage CRadv[A, H] is negligible. Here, CRadv[A, H] is defined to be probability that A can produce a collision, i.e., a pair m0 , m1 2 M such that m0 6= m1 but H(m0 ) = H(m1 ).
10.6.1
Collision resistance based on DL
Let G be a cyclic group of prime order q generated by g 2 G. We define a hash function Hdl defined over (Zq ⇥ Zq , G). This hash function is parameterized by the group G and the generator g, along with a randomly chosen u 2 G. Thus, the group G, along with the group elements g and u, are chosen once and for all, and together, they define the hash function Hdl . For ↵, 2 Zq , we define Hdl (↵, ) := g ↵ u . Notice that a collision on Hdl consists of ↵, , ↵0 , (↵, ) 6= (↵0 ,
0
0
2 Zq such that 0
0
) and g ↵ u = g ↵ u .
(10.1)
That is, a collision of this type gives us two di↵erent ways to represent the same group element as a power of g times a power of u. The problem of finding a collision of this type is sometimes called the representation problem. Theorem 10.3. The hash function Hdl is collision resistant under the DL assumption. In particular, for every collisionfinding adversary A, there exists a DL adversary B, which is an elementary wrapper around A, such that CRadv[A, Hdl ] = DLadv[B, G]. 0
Proof. Consider a collision as in (10.1). Notice that g ↵ u = g ↵ u g↵
↵0
0
u
= 1,
(10.2) 0
implies (10.3)
0 6= 0. Thus, we have a nontrivial representation of 1 as a power where either ↵ ↵0 6= 0 or of g times a power of u. 0 6= 0. To see this, suppose by way of contradiction that 0 = 0. Then We claim that 0 (10.3) implies g ↵ ↵ = 1, and since g is a generator for G, this would mean ↵ ↵0 = 0. Thus, we 0 = 0, which is a contradiction. would have ↵ ↵0 = 0 and 0 6= 0 and q is prime, it follows that 0 has a multiplicative inverse in Z , which Since q we can in fact efficiently compute (see Section ??). So we can rewrite (10.3) as
u = g (↵
0
↵)/(
400
0)
,
which means Dlogg (u) = (↵0
↵)/(
0
).
(10.4)
So we use the given collisionfinding adversary A to build a DL adversary B as follows. When B receives its challenge u 2 G from its DLchallenger, B runs A using Hdl , which is defined using G, g, and the given u. If A produces a collision as in (10.1), adversary B computes and outputs Dlogg (u) as in (10.4). By the above discussion, (10.2) is clear. 2 The function Hdl : Zq ⇥ Zq ! G maps from a message space of size q 2 to a digest space of size q. The good news is that the message space is larger than the digest space, and so the hash function actually compresses. The bad news is that the set of encodings of G may be much larger than the set G itself. Indeed, if G is constructed as recommended in Section 10.4 as a subset of Z⇤p , then elements of G are encoded as 2048bit strings, even though the group G itself has order ⇡ 2256 . So if we replace the set G by the set of encodings, the hash function Hdl is not compressing at all. This problem can be avoided by using other types of groups with more compact encodings, such as elliptic curve groups (see Chapter 16). See also Exercise 10.17 and Exercise 10.18.
10.6.2
Collision resistance based on RSA
We shall work with an RSA encryption exponent e that is a prime. For this application, the bigger e is, the more compression we get. Let Ie := {0, . . . , e 1}. Let n be an RSA modulus, generated as in Section 10.3 using an appropriate length parameter `. We also choose a random y 2 Z⇤n . The values e, n, and y are chosen once and for all, and together they determine a hash function Hrsa defined over (Z⇤n ⇥ Ie , Z⇤n ) as follows: for a 2 Z⇤n and b 2 Ie , we define Hrsa (a, b) := ae y b . We will show that Hrsa is collision resistant under the RSA assumption. Note that Hrsa can be used directly as a compression function in the MerkleDamg˚ ard paradigm (see Section 8.4) to build a collisionresistant hash function for arbitrarily large message spaces. In applying Theorem 8.3, we would take X = Z⇤n and Y = {0, 1}blog2 ec . To analyze Hrsa , we will need a couple of technical results. The first result simply says that in the RSA attack game, it is no easier to compute an eth root of a random element of Z⇤n than it is to compute an eth root of a random element of Zn . To make this precise, suppose that we modify Attack Game 10.3 so that the challenger chooses x R Z⇤n , and keep everything else the same. Note that since x is uniformly distributed over Z⇤n , the value y := xe is also uniformly distributed over Z⇤n . Denote by uRSAadv[A, `, e] the adversary A’s advantage in this modified attack game. Theorem 10.4. Let ` > 2 and odd e > 2 be integers. For every adversary A, there exists an an adversary B, which is an elementary wrapper around A, such that uRSAadv[A, `, e] RSAadv[B, `, e]. Proof. Let A be a given adversary. Here is how B works. Adversary B receives a random element y 2 Zn . If y 2 Z⇤n , then B gives y to A and outputs whatever A outputs. Otherwise, B computes an eth root x of y as follows. If y = 0, B sets x := 0; otherwise, by computing the GCD of y and n, B can factor n, compute the RSA decryption exponent d, and then compute x := y d . Let W be the event that B succeeds. We have / Z⇤n ] Pr[y 2 / Z⇤n ]. Pr[W ] = Pr[W  y 2 Z⇤n ] Pr[y 2 Z⇤n ] + Pr[W  y 2 401
The result follows from the observations that Pr[W  y 2 Z⇤n ] = uRSAadv[A, `, e] and Pr[W  y 2 / Z⇤n ] = 1
uRSAadv[A, `, e].
2
The above theorem shows that the standard RSA assumption implies a variant RSA assumption, where the preimage is chosen at random from from Z⇤n , rather than Zn . In Exercise 10.22, you are to show the converse, that is, that this variant RSA assumption implies the standard RSA assumption. We also need the following technical result, which says that given y 2 Z⇤n , along with an integer f that is relatively prime to e, and an eth root of y f , we can easily compute an eth root of y itself. Just to get a feeling for the result, suppose e = 3 and f = 2. We have w 2 Zn⇤ such that w3 = y 2 . We want to compute x 2 Z⇤n such that x3 = y. If we set x := (y/w), then we have x3 = y 3 /w3 = y 3 /y 2 = y. Theorem 10.5 (Shamir’s trick). There is an efficient algorithm that takes as input n, e, f, w, y, where n is a positive integer, e and f are relatively prime integers, and w and y are elements of Z⇤n that satisfy we = y f , and outputs x 2 Z⇤n such that xe = y. Proof. Using the extended Euclidean algorithm (see Section ??), we compute integers s and t such that es + f t = gcd(e, f ), and output x := y s wt . If gcd(e, f ) = 1 and we = y f , then xe = (y s wt )e = y es wet = y es y f t = y es+f t = y 1 = y.
2
Theorem 10.6. The hash function Hrsa is collision resistant under the RSA assumption. In particular, for every collisionfinding adversary A, there exists an RSA adversary B, which is an elementary wrapper around A, such that CRadv[A, Hrsa ] RSAadv[B, `, e].
(10.5)
Proof. We construct an adversary B 0 that plays the alternative RSA attack game considered in Theorem 10.4. We will show that CRadv[A, Hrsa ] = uRSAadv[B 0 , `, e], and the theorem will the follow from Theorem 10.4. Our RSA adversary B 0 runs as follows. It receives (n, y) from its challenger, where n is an RSA modulus and y is a random element of Z⇤n . The values e, n, y define the hash function Hrsa , and adversary B 0 runs adversary A with this hash function. Suppose that A finds a collision. This is a pair of inputs (a, b) 6= (a0 , b0 ) such that 0
ae y b = (a0 )e y b , which we may rewrite as (a/a0 )e = y b
0
b
.
Using this collision, B 0 will compute an eth root of y. Observe that b0 b 6= 0, since otherwise we would have (a/a0 ) = 1 and hence a = a0 . Also observe that since b b0  < e and e is prime, we must have gcd(e, b b0 ) = 1. So now we simply apply Theorem 10.5 with n, e, and y as given, and w := a/a0 and f := b0 b. 2 402
Adversary
Alice
↵
R
Zq
u
v
ka
(v 0 )↵ = g ↵
0
g↵
g
↵0
R
0
R
0
Zq Zq
Bob
u0
g↵
v
g
0
0
ka , kb
R
Zq
(u0 ) = g ↵
kb
0
Figure 10.4: Man in the middle attack
10.7
Attacks on the anonymous DiffieHellman protocol
The DiffieHellman key exchange is secure against a passive eavesdropper. Usually, however, an attacker capable of eavesdropping on traffic is also able to inject its own messages. The protocol completely falls apart in the presence of an active adversary who controls the network. The main reason is the lack of authentication. Alice sets up a shared secret, but she has no idea with whom the secret is shared. The same holds for Bob. An active attacker can abuse this to expose all traffic between Alice and Bob. The attack, called a man in the middle attack, works against any key exchange protocol that does not include authentication. It works as follows (see Fig. 10.4): • Alice sends (g, g ↵ ) to Bob. The attacker blocks this message from reaching Bob. He picks a 0 random ↵0 R Zn and sends (g, g ↵ ) to Bob. • Bob responds with g . The attacker blocks this message from reaching Alice. He picks a 0 random 0 R Zn and sends g to Alice. • Now Alice computes the key kA := g ↵ both kA and kB .
0
0
and Bob computes kB := g ↵ . The attacker knows
At this point Alice thinks kA is a secret key shared with Bob and will use kA to encrypt messages to him. Similarly for Bob with his key kB . The attacker can act as a proxy between the two. He E(kB , mi ) and forwards intercepts each message ci := E(kA , mi ) from Alice, reencrypts it as c0i c0i to Bob. He also reencrypts messages from Bob to Alice. The communication channel works properly for both parties and they have no idea that this proxying is taking place. The attacker, however, sees all plaintexts in the clear. This generic attack explains why we view key exchange secure against eavesdropping as a toy problem. Protocols secure in this model can completely fall apart once the adversary can tamper with traffic. We will come back to this problem in Chapter 20, where we design protocols secure against active attackers.
403
Alice
Bob Puzzles P1 , . . . , PL j D(k, c2 )
`
k
R
{1, . . . , `}
Pj = (c1 , c2 , c3 )
s`
k
s`
Figure 10.5: Merkle puzzles protocol
10.8
Merkle puzzles: a partial solution to key exchange using block ciphers
Can we build a secure key exchange protocol using symmetrickey primitives? The answer is yes, but the resulting protocol is very inefficient. We show how to do key exchange using a block cipher E = (E, D) defined over (K, M). Alice and Bob want to generate a random s 2 M that is unknown to the adversary. They use a protocol called Merkle puzzles (due to the same Merkle from the MerkleDamg˚ ard hashing paradigm). The protocol, shown in Fig. 10.5, works as follows: Protocol 10.1 (Merkle puzzles). 1. Alice picks random triples (ki , si ) R K ⇥ M for i = 1, . . . , L. We will determine the optimal value for L later. She constructs L puzzles where puzzle Pi0 is defined as: Pi0 =
E(ki , si ), E(ki , i), E(ki , 0)
Next, she sends the L puzzles in a random order to Bob. That is, she picks a random 0 0 permutation ⇡ R Perms[{1, . . . , L}] and sends (P1 , . . . , PL ) := (P⇡(1) , . . . , P⇡(L) ) to Bob. 2. Bob picks a random puzzle Pj = (c1 , c2 , c3 ) where j R {1, . . . , L}. He solves the puzzle by brute force, by trying all keys k 2 K until he finds one such that D(k, c3 ) = 0.
(10.6)
In the unlikely event that Bob finds two di↵erent keys that satisfy (10.6), he indicates to Alice that the protocol failed, and they start over. Otherwise, Bob computes ` D(k, c2 ) and s D(k, c1 ), and sends ` back to Alice. 3. Alice locates puzzle P`0 and sets s
s` . Both parties now know the shared secret s 2 M.
Clearly, when the protocol terminates successfully, both parties agree on the same secret s 2 M. Moreover, when M is much larger than K, the protocol is very likely to terminate successfully, because under these conditions (10.6) is likely to have a unique solution. The work for each party in this protocol is as follows: Alice’s work = O(L),
Bob’s work = O(K). 404
Hence, to make the workload for the two parties about the same we need to set L ⇡ K. Either way, the size of L and K needs to be within reason so that both parties can perform the computation in a reasonable time. For example, one can set L ⇡ K ⇡ 230 . When using AES one can force K to have size 230 by fixing the 98 most significant bits of the key to zero. Security. The adversary sees the protocol transcript which includes all the puzzles and the quantity ` sent by Bob. Since the adversary does not know which puzzle Bob picked, intuitively, he needs to solve all puzzles until he finds puzzle P` . Thus, to recover s 2 M the adversary must solve L puzzles each one taking O(K) time to solve. Overall, the adversary must spend time O(LK). One can make this argument precise, by modeling the block cipher E as an ideal cipher, as we did in Section 4.7. We can assume that K is polybounded, and that M is superpoly. Then the analysis shows that if the adversary makes at most Q queries to the ideal cipher, then its probability Q of learning the secret s 2 M is bounded by approximately LK . Working out the complete proof and the exact bound is a good exercise in working with the ideal cipher model. Performance. Suppose we set L ⇡ K. Then the adversary must spend time O(L2 ) to break the protocol, while each participant spends time O(L). Hence, there is a quadratic gap between the work of the participants and the work to break the protocol. Technically speaking, this doesn’t satisfy our definitions of security — with constant work the adversary has advantage about 1/L2 which is nonnegligible. Even worse, in practice one would have to make L extremely large to have a reasonable level of security against a determined attacker. The resulting protocol is then very inefficient. Nevertheless, the Merkle puzzles protocol is very elegant and shows what can be done using block ciphers alone. As the story goes, Merkle came up with this clever protocol while taking a seminar as an undergraduate student at Berkeley. The professor gave the students the option of submitting a research paper instead of taking the final exam. Merkle submitted his key exchange protocol as the research project. These ideas, however, were too far out and the professor rejected the paper. Merkle still had to take the final exam. Subsequently, for his Ph.D. work, Merkle chose to move to a di↵erent school to work with Martin Hellman. It is natural to ask if a better key exchange protocol, based on block ciphers, can achieve better than quadratic separation between the participants and the adversary. Unfortunately, a result by Impagliazzo and Rudich [57] suggests that one cannot achieve better separation using block ciphers alone.
10.9
Fun application: Pedersen commitments
To be written.
10.10
Notes
Citations to the literature to be added.
405
10.11
Exercises
10.1 (Computationally unbounded adversaries). Show that an anonymous key exchange protocol P (as in Definition 10.1) cannot be secure against a computationally unbounded adversary. This explains why all protocols in this chapter must rely on computational assumptions. 10.2 (DDH PRG). Let G be a cyclic group of prime order q generated by g 2 G. Consider the following PRG defined over (Z2q , G3 ): G(↵, ) := (g ↵ , g , g ↵ ). Show that G is a secure PRG assuming DDH holds in G. 10.3 (The NaorReingold PRF). Let G be a cyclic group of prime order q generated by g 2 G. , {0, 1}n , G is secure assuming DDH holds Let us show that the following PRF defined over Zn+1 q in G: ⌘ ⇣ x1 xn FNR (↵0 , ↵1 , . . . , ↵n ), (x1 , . . . , xn ) := g (↵0 ·↵1 ···↵n )
This secure PRF is called the NaorReingold PRF.
(a) We prove security of FNR using Exercise 4.18. First, show that FNR is an augmented tree construction constructed from the PRG: GNR (↵, g ) := (g , g ↵ ). (b) Second, show that GNR satisfies the hypothesis of Exercise 4.18 part (b), assuming DDH holds in G. Use the result of Exercise 10.10. Security of FNR now follows from Exercise 4.18 part (b). Discussion: See Exercise 11.1 for a simpler PRF from the DDH assumption, but in the random oracle model. 10.4 (Random selfreduction for CDH (I)). Consider a specific cyclic group G of prime order q generated by g 2 G. For u = g ↵ 2 G and v = g 2 G, define [u, v] = g ↵ , which is the solution instance (u, v) of the CDH problem. Consider the randomized mapping from G2 to G2 that sends (u, v) to (˜ u, v), where ⇢ R Zq , u ˜ g ⇢ u. Show that (a) u ˜ is uniformly distributed over G; (b) [˜ u, v] = [u, v] · v ⇢ . 10.5 (Random selfreduction for CDH (II)). Continuing with the previous exercise, suppose A is an efficient algorithm that solves the CDH problem with success probability ✏ on random inputs. That is, if u, v 2 G are chosen at random, then Pr[A(u, v) = [u, v]] = ✏, where the probability is over the random choice of u and v, as well as any random choices made by A. Using A, construct an efficient algorithm B that solves the CDH problem with success probability ✏ for all inputs. More precisely, for all u, , v 2 G, we have Pr[B(u, v) = [u, v]] = ✏, where the probability is now only over the random choices made by B.
Remark: If we iterate B on the same input (u, v) many times, say nd1/✏e times for some n, at least one of these iterations will output the correct result [u, v] with probability 1 (1 ✏)nd1/✏e 406
1 exp( n). Unfortunately, assuming the DDH is true, we will have no way of knowing which of these outputs is the correct result. 10.6 (An alternative DDH characterization). Let G by a cyclic group of prime order q generated by g 2 G. Let P be the uniform distribution over G3 . Let Pdh be the uniform distribution over the set of all DHtriples (g ↵ , g , g ↵ ). Let Pndh be the uniform distribution over the set of all nonDHtriples (g ↵ , g , g ), 6= ↵ . (a) Show that the statistical distance between P and Pndh is 1/q. (b) Using part (a), deduce that under the DDH assumption, the distributions Pdh and Pndh are computationally indistinguishable. 10.7 (Random selfreduction for DDH (I)). Consider a specific cyclic group G of prime order q generated by g 2 G. Let DH be the set of all DHtriples, i.e., DH := {(g ↵ , g , g ↵ ) 2 G3 : ↵,
2 Zq }.
For fixed u 2 G, and let Tu be the subset of G3 whose first coordinate is u. Consider the randomized mapping from G3 to G3 that sends (u, v, w) to (u, v ⇤ , w⇤ ), where R
Zq , ⌧
R
Zq , v ⇤
g v ⌧ , w⇤
u w⌧ .
Prove the following: (a) if (u, v, w) 2 DH, then (u, v ⇤ , w⇤ ) is uniformly distributed over DH \ Tu ; (b) if (u, v, w) 2 / DH, then (u, v ⇤ , w⇤ ) is uniformly distributed over Tu . 10.8 (Random selfreduction for DDH (II)). Continuing with the previous exercise, consider u, v, w), ˜ where the randomized mapping from G3 to G3 that sends (u, v, w) to (˜ ⇢
R
Zq , u ˜
g ⇢ u, w ˜
v ⇢ w.
Prove the following: (a) u ˜ is uniformly distributed over G; (b) (u, v, w) 2 DH () (˜ u, v, w) ˜ 2 DH; (c) if we apply the randomized mapping from the previous exercise to (˜ u, v, w), ˜ obtaining the ⇤ ⇤ ˜ ), then we have triple (˜ u, v , w ˜ ⇤ ) is uniformly distributed over DH; • if (u, v, w) 2 DH, then (˜ u, v ⇤ , w
• if (u, v, w) 2 / DH, then (˜ u, v ⇤ , w ˜ ⇤ ) is uniformly distributed over G3 .
10.9 (Random selfreduction for DDH (III)). Continuing with the previous exercise, prove the following. Suppose A is an efficient algorithm that takes as input three group elements and outputs a bit, and which satisfies the following property: if ↵, , 2 Zq are chosen at random, then Pr[A(g ↵ , g , g ↵ ) = 1]
Pr[A(g ↵ , g , g ) = 1] = ✏,
407
where the probability is over the random choice of ↵, , , as well as any random choices made by A. Assuming that 1/✏ is polybounded, show how to use A to build an efficient algorithm B that for all inputs (u, v, w) correctly decides whether or not (u, v, w) 2 DH with negligible error probability. That is, adversary B may output an incorrect answer, but for all inputs, the probability that its answer is incorrect should be negligible. Hint: Use a Cherno↵ bound. 10.10 (MultiDDH). Let G be a cyclic group of prime order q generated by g 2 G. Let n m be positive integers. Define the following two distributions over Gn·m+n+m : D:
g ↵i (i = 1, . . . , n),
R:
g ↵i (i = 1, . . . , n),
g ↵i
j
g
j
(j = 1, . . . , m)
(i = 1, . . . , n, j = 1, . . . , m),
and g
ij
g
j
(j = 1, . . . , m)
(i = 1, . . . , n, j = 1, . . . , m).
where the ↵i ’s, j ’s, and ij ’s are uniformly and independently distributed over Zq . Show that under the DDH assumption, D and R are computationally indistinguishable (as in Definition 3.4). In particular, show that for every adversary A that distinguishes D and R, there exists a DDH adversary B (which is an elementary wrapper around A) such that Distadv[A, D, R] n · (1/q + DDHadv[B, G]). Hint: First give a proof for the case n = 1 using the results of Exercise 10.6 and Exercise 10.7, and then generalize to arbitrary n using a hybrid argument. , Gn·m+n+m ), with a Discussion: This result gives us a DDHbased PRG G defined over (Zn+m q nice expansion rate, given by ⇣ ⌘ ⇣ ⌘ ↵i n ↵i j j m := G {↵i }ni=1 , { j }m } , {g } , {g } {g i=1,...,n j=1 i=1 j=1 j=1,...,m
10.11 (Matrix DDH). Let G be a cyclic group of prime order q generated by g 2 G. Let n and m be positive integers, and assume n m. For A = (↵ij ) 2 Zn⇥m (i.e., A is an n ⇥ m matrix with q entries in Zq ), let g A be the n ⇥ m matrix whose entry at row i column j is the group element g ↵ij . For k = 1, . . . , n, define the random variable R(k) to be a random matrix uniformly distributed over all n ⇥ m matrices over Zq of of rank k. Let 1 k1 < k2 n. Show that g R(k1 ) and g R(k2 ) are computationally indistinguishable under the DDH. In particular, show that for every adversary A that distinguishes g R(k1 ) and g R(k2 ) there exists a DDH adversary B (which is an elementary wrapper around A) such that Distadv[A, g R(k1 ) , g R(k2 ) ] (k2
k1 ) · (1/q + DDHadv[B, G]).
is a fixed matrix of rank k, and if U 2 Zn⇥n and V 2 Zm⇥m Hint: Use the fact that if A 2 Zn⇥m q q q n⇥m are a random invertible matrices, then the matrix U AV 2 Zq is uniformly distributed over all n ⇥ m matrices of rank k. You might also try to prove this fact, which is not too hard. Discussion: For k1 = 1 and k2 = n, this result implies a closely related, but slightly weaker form of Exercise 10.10. In this sense, this exercise is a generalization of Exercise 10.10. 408
10.12 (A trapdoor test). Consider a specific cyclic group G of prime order q generated by g 2 G. Let u 2 G and f : G ! G3 . Now set R
Zq , ⌧
R
Zq , u ¯
g u⌧ , (v, w, w) ¯
f (¯ u).
Let S be the the event that (u, v, w) and (¯ u, v, w) ¯ are both DHtriples. Let T be the event that ⌧ w ¯ = v w . Show that: (a) u ¯ is uniformly distributed over G; (b) Pr[S ^ ¬T ] = 0; (c) Pr[¬S ^ T ] 1/q.
Remark: This result gives us a kind of trapdoor test. Suppose a group element u 2 G is given (it could be chosen at random or adversarially chosen). Then we can generate a random element u ¯ and a “trapdoor” ( , ⌧ ). Using this trapdoor, given group elements v, w, w ¯ 2 G (possibly adversarially chosen in a way that depends on u ¯), we can reliably test if (u, v, w) and (¯ u, v, w) ¯ are both DHu), and even though we cannot tell triples, even though we do not know either Dlogg (u) or Dlogg (¯ whether (u, v, w) and (¯ u, v, w) ¯ are individually DHtriples. This rather technical result has several nice applications, one of which is developed in the following exercise. 10.13 (A CDH selfcorrector). Consider a specific cyclic group G of prime order q generated by g 2 G. Let A be an efficient algorithm with the following property: if ↵, 2 Zq are chosen at random, then Pr[A(g ↵ , g ) = g ↵ ] = ✏. Here, the probability is over the random choice of ↵ and , as well as any random choices made by A. Assuming 1/✏ is polybounded and G is superpoly, show how to use A to build an efficient algorithm B that solves the CDH problem on all inputs with negligible error probability; that is, on every input (g ↵ , g ), algorithm B outputs a single group element w, and w 6= g ↵ with negligible probability (and this probability is just over the random choices made by B). Here is a highlevel sketch of how B might work on input (u, v). somehow choose u ¯2G ¯ of group elements somehow use A to generate lists L, L ¯ for each w in L and each w ¯ in L do if (u, v, w) and (¯ u, v, w) ¯ are both DHtriples then output w and halt output an arbitrary group element As stated, this algorithm is not fully specified. Nevertheless, you can use this rough outline, combined with the CDH random self reduction in Exercise 10.4 and the trapdoor test in Exercise 10.12, to prove the desired result. For the next problem, we need the following notions from complexity theory: • We say problem A is deterministic polytime reducible to problem B if there exists a deterministic algorithm R for solving problem A on all inputs that makes calls to a subroutine that solves problem B on all inputs, where the running time of R (not including the running time for the subroutine for B) is polynomial in the input length. 409
• We say that A and B are deterministic polytime equivalent if A is deterministic polytime reducible to B and B is deterministic polytime reducible to A. 10.14 (Problems equivalent to DH). Consider a specific cyclic group G of prime order q generated by g 2 G. Show that the following problems are deterministic polytime equivalent: (a) Given g ↵ and g , compute g ↵ (this is just the DiffieHellman problem). 2
(b) Given g ↵ , compute g (↵ ) . (c) Given g ↵ with ↵ 6= 0, compute g 1/↵ . (d) Given g ↵ and g with
6= 0, compute g ↵/ .
Note that all problem instances are defined with respect to the same group G and generator g 2 G. 10.15 (System parameters). In formulating the discretelog Attack Game 10.4, we assume that the description of G, including g 2 G and q, is a system parameter that is generated once and for all at system setup time and shared by all parties involved. This parameter may be generated via some randomized process, in which case the advantage ✏ = DLadv[A, G] is a probability over the choice of system parameter, as well as the random choice of ↵ 2 Zq made by the challenger and any random choices made by adversary. So we can think of the system parameter as a random variable ⇤, and for any specific system parameter ⇤0 , we can consider the corresponding conditional advantage ✏(⇤0 ) given that ⇤ = ⇤0 , which is a probability just over the random choice of ↵ 2 Zq made by the challenger and any random choices made by adversary. Let us call ⇤0 a “vulnerable” parameter if ✏(⇤0 ) ✏/2. (a) Prove that the probability that ⇤ is vulnerable is at least ✏/2. Note that even if an adversary breaks the DL with respect to a randomly generated system parameter, there could be many particular system parameters for which the adversary cannot or will not break the DL (it is helpful to imagine an adversary that is all powerful yet capricious, who simply refuses to break the DL for certain groups and generators which he finds distasteful). This result says, however, that there is still a nonnegligible fraction of vulnerable system parameters for which the adversary breaks the DL. (b) State and prove an analogous result for the CDH problem. (c) State and prove an analogous result for the DDH problem. 10.16 (Choice of generators). In formulating the DL, CDH, and DDH assumptions, we work with a cyclic group G of prime order q generated by g 2 G. We do not specify how the generator g is chosen. Indeed, it may be desirable to choose a specific g that allows for more efficient implementations. Conceivably, such a g could be a “weak” generator that makes it easier for an adversary to break the DL, CDH, or DDH assumptions. So to be on the safe side, we might insist that the generator g is uniformly distributed over G\{1}. If we do this, we obtain new assumptions, which we call the rDL, rCDH, and rDDH assumptions. Show that: (a) the rDL and DL assumptions are equivalent; (b) the rCDH and CDH assumptions are equivalent;
410
(c) the DDH assumption implies the rDDH assumption. Hint: To start with, you might first consider the setting where we are working with a specific group, then generalize your result to incorporate all the aspects of the asymptotic attack game (see Section 10.5.2), including the security parameter and the system parameter (where the group is selected at system setup time). Remark: The rDDH assumption is not known to imply the DDH assumption, so for applications that use the DDH assumption, it seems safest to work with a random generator. 10.17 (Collision resistance from discretelog). Let G be a cyclic group of prime order q generated by g 2 G. Let n be a polybounded parameter. We define a hash function H defined over (Znq , G). The hash function is parameterized by the group G and n randomly chosen group elements g1 , . . . , gn 2 G. For (↵1 , . . . , ↵n ) 2 Znq , we define H(↵1 , . . . , ↵n ) := g1↵1 · · · gn↵n . Prove that H is collision resistant under the DL assumption for G. In particular, show that for every collisionfinding adversary A, there exists a DL adversary B, which is an elementary wrapper around A, such that CRadv[A, H] DLadv[B, G] + 1/q. 10.18 (Collision resistance in Z⇤p ). This exercise asks you to prove that the hash function presented in Section 8.5.1 is collision resistant under an appropriate DL assumption. Let us define things a bit more precisely. Let p be a large prime such that q := (p 1)/2 is also prime. The prime q is called a Sophie Germain prime, and p is sometimes called a “strong” prime. Such primes are often very convenient to use in cryptography. Suppose x is a randomly chosen integer in the range [2, q] and y is a randomly chosen integer in the range [1, q]. These parameters define a hash function H that takes as input two integers in [1, q] and outputs an integer in [1, q], as specified in (8.3). Let G be the subgroup of order q in Z⇤p , and consider the DL assumption for G with respect to a randomly chosen generator. Show that H is collision resistant under this DL assumption. Hint: Use the fact that and that the map that sends ↵ 2 Z⇤p to ↵2 2 Z⇤p is a group homomorphism with image G and kernel ±1; also use the fact that there is an efficient algorithm for taking square roots in Z⇤p . 10.19 (A broken CRHF). Consider the following variation of the hash construction in the previous exercise. Let p be a large prime such that q := (p 1)/2 is also prime. Let x and y be randomly chosen integers in the range [2, p 2] (so neither can be ±1 (mod p)). These parameters define a hash function H that takes as input two integers in [1, p 1] and outputs an integer in [1, p 1], as follows: H(a, b) := xa y b mod p. Give an efficient, deterministic algorithm that takes as input p, x, y as above, and computes a collision on the corresponding H. Your algorithm should work for all inputs p, x, y. 10.20 (DDH is easy in groups of even order). We have restricted the DL, CDH, and DDH assumptions to prime order groups G. Consider the DDH assumption for a cyclic group G of even order q with generator g 2 G. Except for dropping the restriction that q is prime, the attack game is identical to Attack Game 10.6. Give an efficient adversary that has advantage 1/2 in solving the DDH for G. 411
Remark: For a prime p > 2, the group Z⇤p is a cyclic group of even order p 1. This exercise shows that the DDH assumption is false in this group. Exercise 10.19 gives another reason to restrict ourselves to groups of prime order. 10.21 (RSA variant (I)). Let n be an RSA modulus generated by RSAGen(`, e). Let X and X ⇤ be random variables, where X is uniformly distributed over Zn and X ⇤ is uniformly distributed over Z⇤n . Show that the statistical distance [X, X ⇤ ] is less than 2 (` 2) . 10.22 (RSA variant (II)). In Theorem 10.4, we considered a variant of the RSA assumption where the challenger chooses the preimage x at random from Z⇤n , rather than Zn . That theorem showed that the standard RSA assumption implies this variant RSA assumption. In this exercise, you are to show the converse. In particular, show that RSAadv[A, `, e] uRSAadv[B, `, e] + 2 (` 2) for every adversary A. Hint: Use the result of the previous exercise. 10.23 (A proper trapdoor permutation scheme based on RSA). As discussed in Section 10.3, our RSAbased trapdoor permutation scheme does not quite satisfy our definitions, simply because the domain on which it acts varies with the public key. This exercise shows one way to patch things up. Let ` and e be parameters used for RSA key generation, and let G be the key generation algorithm, which outputs a pair (pk , sk ). Recall that pk = (n, e), where n is an RSA modulus, which is the product of two `bit primes, and e is the encryption exponent. The secret key is sk = (n, d), where d is the decryption exponent corresponding to the encryption exponent e. Choose a parameter L that is a substantially larger than 2`, so that n/2L is negligible. Let X be the set of integers in the range [0, 2L ). We shall present a trapdoor permutation scheme (G, F ⇤ , I ⇤ ), defined over X . The function F ⇤ takes two inputs: a public key pk as above and an integer x 2 X , and outputs an integer y 2 X , computed as follows. Divide x by n to obtain the integer quotient Q and remainder R, so that x = nQ + R and 0 R < n. If Q > 2L /n 1, then set S := R; otherwise, set S := Re mod n. Finally, set y := nQ + S. (a) Show that F ⇤ (pk , ·) is a permutation on X , and give an efficient inversion function I ⇤ that satisfies I ⇤ (sk , F ⇤ (pk , x)) = x for all x 2 X . (b) Show under the RSA assumption, (G, F ⇤ , I ⇤ ) is oneway. 10.24 (Random selfreduction for RSA). Suppose we run (n, d) R RSAGen(`, e). There could be “weak” RSA moduli n for which an adversary can break the the RSA assumption with some probability ✏. More precisely, suppose that there is an efficient algorithm A such that for any such “weak” modulus n, if x 2 Z⇤n is chosen at random, then Pr[A(xe ) = x] ✏, where the probability is over the random choice of x, as well as any random choices made by A. Using A, construct an efficient algorithm B such that for every “weak” modulus n, and every x 2 Zn , we have Pr[A(xe ) = x] ✏, where the probability is now only over the random choices made by B. Hint: Use the randomized mapping from Z⇤n to Z⇤n that sends y to y˜, where r Show that for every y 2 Z⇤n , the value y˜ is uniformly distributed over Z⇤n .
R
Z⇤n , y˜
re y.
10.25 (nproduct CDH). Let G be a cyclic group of prime order q generated by g 2 G. The following attack game defines the nproduct CDH problem (here, n is a polybounded parameter, not necessarily constant). The challenger begins by choosing ↵i R Zq for i = 1, . . . , n. The adversary then makes a sequence of queries. In each query, the adversary submits proper a subset 412
of indices S ( {1, . . . , n}, and the challenger responds with g
Q
i2S
↵i
.
The adversary wins the game if he outputs g ↵1 ···↵n . We relate the hardness of solving the nproduct CDH problem to another problem, called the npower CDH problem. In the attack game for this problem, the challenger begins by choosing ↵ R Z⇤q , and gives g, g ↵ , . . . , g ↵
n 1
n
to the adversary. The adversary wins the game if he outputs g ↵ . Show that if there is an efficient adversary A that breaks nproduct CDH with nonnegligible probability, then there is an efficient adversary B that breaks npower CDH with nonnegligible probability. 10.26 (Trapdoor collison resistance). Let us show that the collision resistant hash functions Hdl and Hrsa , presented in Section 10.6, are trapdoor collision resistant. (a) Recall that Hdl is defined as Hdl (↵, ) := g ↵ u 2 G, where g and u are parameters chosen at setup. Show that anyone who knows the discretelog of u base g (the trapdoor), can break the 2ndpreimage resistance of Hdl . That is, given (↵, ) as input, along with the trapdoor, one can efficiently compute (↵0 , 0 ) 6= (↵, ) such that Hdl (↵0 , 0 ) = Hdl (↵, ). (b) Recall that Hrsa is defined as Hrsa (a, b) := ae y b 2 Zn , where n, e and y are parameters chosen at setup. Show that anyone who knows the eth root of y in Zn (the trapdoor), can break the 2ndpreimage resistance of Hrsa . (c) Continuing with part (b), show that anyone who knows the factorization of n (the trapdoor), can invert Hrsa . That is, given z 2 Zn as input, one can find (a, b) such that Hrsa (a, b) = z. Discussion: Part (c) shows that the factorization of n is a “stronger” trapdoor for Hrsa than the eth root of y. The latter only breaks 2ndpreimage resistance of Hrsa , where as the former enables complete inversion. Both trapdoors break collision resistance.
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Chapter 11
Public key encryption In this chapter, we consider again the basic problem of encryption. As a motivating example, suppose Alice wants to send Bob an encrypted email message, even though the two of them do not share a secret key (nor do they share a secret key with some common third party). Surprisingly, this can be done using a technology called publickey encryption. The basic idea of publickey encryption is that the receiver, Bob in this case, runs a key generation algorithm G, obtaining a pair of keys: (pk , sk )
R
G().
The key pk is Bob’s public key, and sk is Bob’s secret key. As their names imply, Bob should keep sk secret, but may publicize pk . To send Bob an encrypted email message, Alice needs two things: Bob’s email address, and Bob’s public key pk . How Alice reliably obtains this information is a topic we shall explore later in Section 13.8. For the moment, one might imagine that this information is placed by Bob in some kind of public directory to which Alice has readaccess. So let us assume now that Alice has Bob’s email address and public key pk . To send Bob an encryption of her email message m, she computes the ciphertext c
R
E(pk , m).
She then sends c to Bob, using his email address. At some point later, Bob receives the ciphertext c, and decrypts it, using his secret key: m
D(sk , c).
Publickey encryption is sometimes called asymmetric encryption to denote the fact that the encryptor uses one key, pk , and the decryptor uses a di↵erent key, sk . This is in contrast with symmetric encryption, discussed in Part 1, where both the encryptor and decryptor use the same key. A few points deserve further discussion: • Once Alice obtains Bob’s public key, the only interaction between Alice and Bob is the actual transmission of the ciphertext from Alice to Bob: no further interaction is required. In fact, we chose encrypted email as our example problem precisely to highlight this feature, as email delivery protocols do not allow any interaction beyond delivery of the message. 414
• As we will discuss later, the same public key may be used many times. Thus, once Alice obtains Bob’s public key, she may send him encrypted messages as often as she likes. Moreover, other users besides Alice may send Bob encrypted messages using the same public key pk . • As already mentioned, Bob may publicize his public key pk . Obviously, for any secure publickey encryption scheme, it must be hard to compute sk from pk , since anyone can decrypt using sk .
11.1
Two further example applications
Publickey encryption is used in many realworld settings. We give two more examples.
11.1.1
Sharing encrypted files
In many modern file systems, a user can store encrypted files to which other users have read access: the owner of the file can selectively allow others to read the unencrypted contents of the file. This is done using a combination of publickey encryption and an ordinary, symmetric cipher. Here is how it works. Alice encrypts a file f under a key k, using an ordinary, symmetric cipher. The resulting ciphertext c is stored on the file system. If Alice wants to grant Bob access to the contents of the file, she encrypts k under Bob’s public key; that is, she computes cB R E(pk B , k), where pk B is Bob’s public key. The ciphertext cB is then stored on the file system near the ciphertext c, say, as part of the file header, which also includes file metadata (such as the file name, modification time, and so on). Now when Bob wants to read the file f , he can decrypt cB using his secret key sk B , obtaining k, using which he can decrypt c using the symmetric cipher. Also, so that Alice can read the file herself, she grants access to herself just as she does to Bob, by encrypting k under her own public key pk A . This scheme scales very nicely if Alice wants to grant access to f to a number of users. Only one copy of the encrypted file is stored on the file system, which is good if the file is quite large (such as a video file). For each user that is granted access to the file, only an encryption of the key k is stored in the file header. Each of these ciphertexts is fairly small (on the order of a few hundred bytes), even if the file itself is very big.
11.1.2
Key escrow
Consider a company that deploys an encrypted file system such as the one described above. One day Alice is traveling, but her manager needs to read one of her files to prepare for a meeting with an important client. Unfortunately, the manager is unable to decrypt the file because it is encrypted and Alice is unreachable. Large companies solve this problem using a mechanism called key escrow. The company runs a key escrow server that works as follows: at setup time the key escrow server generates a secret key sk ES and a corresponding public key pk ES . It keeps the secret key to itself and makes the public key available to all employees. When Alice stores the encryption c of a file f under a symmetric key k, she also encrypts k under pk ES , and then stores the resulting ciphertext cES in the file header. Every file created by company employees is encrypted this way. Now, if Alice’s manager later needs access to f and Alice
415
is unreachable, the manager sends cES to the escrow service. The server decrypts cES , obtaining k, and sends k to the manager, who can then use this to decrypt c and obtain f . Publickey encryption makes it possible for the escrow server to remain o✏ine, until someone needs to decrypt an inaccessible file. Also, notice that although the escrow service allows Alice’s manager to read her files, the escrow service itself cannot read Alice’s files, since the escrow service never sees the encryption of the file.
11.2
Basic definitions
We begin by defining the basic syntax and correctness properties of a publickey encryption scheme. Definition 11.1. A publickey encryption scheme E = (G, E, D) is a triple of efficient algorithms: a key generation algorithm G, an encryption algorithm E, a decryption algorithm D. • G is a probabilistic algorithm that is invoked as (pk , sk ) key and sk is called a secret key.
R
G(), where pk is called a public
• E is a probabilistic algorithm that is invoked as c R E(pk , m), where pk is a public key (as output by G), m is a message, and c is a ciphertext. • D is a deterministic algorithm that is invoked as m D(sk , c), where sk is a secret key (as output by G), c is a ciphertext, and m is either a message, or a special reject value (distinct from all messages). • As usual, we require that decryption undoes encryption; specifically, for all possible outputs (pk , sk ) of G, and all messages m, we have Pr[D(sk , E(pk , m) ) = m] = 1. • Messages are assumed to lie in some finite message space M, and ciphertexts in some finite ciphertext space C. We say that E = (G, E, D) is defined over (M, C). We next define the notion of semantic security for a publickey encryption scheme. We stress that this notion of security only models an eavesdropping adversary. We will discuss stronger security properties in the next chapter. Attack Game 11.1 (semantic security). For a given publickey encryption scheme E = (G, E, D), defined over (M, C), and for a given adversary A, we define two experiments. Experiment b
(b = 0, 1):
• The challenger computes (pk , sk )
R
G(), and sends pk to the adversary.
• The adversary computes m0 , m1 2 M, of the same length, and sends them to the challenger. • The challenger computes c
R
E(pk , mb ), and sends c to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. 416
Challenger (Experiment b)
(pk , sk )
c
R
R
A
pk
G()
m0 , m 1 2 M
c
E(pk , mb )
ˆb 2 {0, 1}
Figure 11.1: Experiment b of Attack Game 11.1 If Wb is the event that A outputs 1 in Experiment b, we define A’s advantage with respect to E as SSadv[A, E] := Pr[W0 ] Pr[W1 ] . 2 Note that in the above game, the events W0 and W1 are defined with respect to the probability space determined by the random choices made by the key generation and encryption algorithms, and the random choices made by the adversary. See Fig. 11.1 for a schematic diagram of Attack Game 11.1. Definition 11.2 (semantic security). A publickey encryption scheme E is semantically secure if for all efficient adversaries A, the value SSadv[A, E] is negligible. As discussed in Section 2.3.5, Attack Game 11.1 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A. In this game, we measure A’s bitguessing advantage SSadv⇤ [A, E] as Pr[ˆb = b] 1/2. The general result of Section 2.3.5 (namely, (2.13)) applies here as well: SSadv[A, E] = 2 · SSadv⇤ [A, E]. (11.1)
11.2.1
Mathematical details
We give a more mathematically precise definition of a publickey encryption scheme, using the terminology defined in Section 2.4. Definition 11.3 (publickey encryption scheme). A publickey encryption scheme consists of a three algorithms, G, E, and D, along with two families of spaces with system parameterization P: M = {M ,⇤ } ,⇤ and C = {C ,⇤ } ,⇤ , such that 1. M and C are efficiently recognizable.
417
2. M has an e↵ective length function. 3. Algorithm G is an efficient probabilistic algorithm that on input , ⇤, where 2 Z 1 , ⇤ 2 Supp(P ( )), outputs a pair (pk , sk ), where pk and sk are bit strings whose lengths are always bounded by a polynomial in . 4. Algorithm E is an efficient probabilistic algorithm that on input , ⇤, pk , m, where 2 Z 1 , ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)) for some sk , and m 2 M ,⇤ , always outputs an element of C ,⇤ . 5. Algorithm D is an efficient deterministic algorithm that on input , ⇤, sk , c, where 2 Z 1 , ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)) for some pk , and c 2 C ,⇤ , outputs either an / M ,⇤ . element of M ,⇤ , or a special symbol reject 2 6. For all , ⇤, pk , sk , m, c, where 2 Z 1 , ⇤ 2 Supp(P ( )), (pk , sk ) 2 Supp(G( , ⇤)), k 2 K ,⇤ , m 2 M ,⇤ , and c 2 Supp(E( , ⇤; pk , m)), we have D( , ⇤; sk , c) = m. As usual, the proper interpretation of Attack Game 11.1 is that both challenger and adversary receive as a common input, and that the challenger generates ⇤ and sends this to the adversary before the game proper begins. The advantage is actually a function of , and security means that this is a negligible function of .
11.3
Implications of semantic security
Before constructing semantically secure publickey encryption schemes, we first explore a few consequences of semantic security. We first show that any semantically secure publickey scheme must use a randomized encryption algorithm. We also show that in the publickey setting, semantic security implies CPA security. This was not true for symmetric encryption schemes: the onetime pad is semantically secure, but not CPA secure.
11.3.1
The need for randomized encryption
Let E = (G, E, D) be a semantically secure publickey encryption scheme defined over (M, C) where M 2. We show that the encryption algorithm E must be a randomized, otherwise the scheme cannot be semantically secure. To see why, suppose E is deterministic. Then the following adversary A breaks semantic security of E = (G, E, D): • A receives a public key pk from its challenger. • A chooses two distinct messages m0 and m1 in M and sends them to its challenger. The challenger responds with c := E(pk , mb ) for some b 2 {0, 1}. • A computes c0 := E(pk , m0 ) and outputs 0 if c = c0 . Otherwise, it outputs 1. Because E is deterministic, we know that c = c0 whenever b = 0. Therefore, when b = 0 the adversary always outputs 0. Similarly, when b = 1 it always outputs 1. Therefore SSadv[A, E] = 1 418
showing that E is insecure. This generic attack explains why semantically secure publickey encryption schemes must be randomized. All the schemes we construct in this chapter and the next use randomized encryption. This is quite di↵erent from the symmetric key settings where a deterministic encryption scheme can be semantically secure; for example, the onetime pad.
11.3.2
Semantic security against chosen plaintext attack
Recall that when discussing symmetric ciphers, we introduced two distinct notions of security: semantic security, and semantic security against chosen plaintext attack (or CPA security, for short). We showed that for symmetric ciphers, semantic security does not imply CPA security. However, for publickey encryption schemes, semantic security does imply CPA security. Intuitively, this is because in the publickey setting, the adversary can encrypt any message he likes, without knowledge of any secret key material. The adversary does so using the given public key and never needs to issue encryption queries to the challenger. In contrast, in the symmetric key setting, the adversary cannot encrypt messages on his own. The attack game defining CPA security in the publickey setting is the natural analog of the corresponding game in the symmetric setting (see Attack Game 5.2 in Section 5.3): Attack Game 11.2 (CPA security). For a given publickey encryption scheme E = (G, E, D), defined over (M, C), and for a given adversary A, we define two experiments. Experiment b
(b = 0, 1):
• The challenger computes (pk , sk )
R
G(), and sends pk to the adversary.
• The adversary submits a sequence of queries to the challenger.
For i = 1, 2, . . . , the ith query is a pair of messages, mi0 , mi1 2 M, of the same length. The challenger computes ci
R
E(pk , mib ), and sends ci to the adversary.
• The adversary outputs a bit ˆb 2 {0, 1}. If Wb is the event that A outputs 1 in Experiment b, then we define A’s advantage with respect to E as CPAadv[A, E] := Pr[W0 ] Pr[W1 ] . 2 Definition 11.4 (CPA security). A publickey encryption scheme E is called semantically secure against chosen plaintext attack, or simply CPA secure, if for all efficient adversaries A, the value CPAadv[A, E] is negligible. Theorem 11.1. If a publickey encryption scheme E is semantically secure, then it is also CPA secure. In particular, for every CPA adversary A that plays Attack Game 11.2 with respect to E, and which makes at most Q queries to its challenger, there exists an SS adversary B, where B is an elementary wrapper around A, such that CPAadv[A, E] = Q · SSadv[B, E].
419
Proof. The proof is a straightforward hybrid argument, and is very similar to the proof of Theorem 5.1. Suppose E = (G, E, D) is defined over (M, C). Let A be a CPA adversary that plays Attack Game 11.2 with respect to E, and which makes at most Q queries to its challenger. We describe the relevant hybrid games. For j = 0, . . . , Q, Hybrid j is played between A and a challenger who works as follows: (pk , sk ) R G() Send pk to A Upon receiving if i > j then else send ci to
the ith query (mi0 , mi1 ) 2 M2 from A do: ci ci A.
R R
E(pk , mi0 ) E(pk , mi1 )
Put another way, the challenger in Hybrid j encrypts m11 , . . . , mj1 ,
m(j+1)0 , . . . , mQ0 ,
As usual, we define pj to be the probability that A outputs 1 in Hybrid j. Clearly, CPAadv[A, E] = pQ
p0 .
Next, we define an appropriate adversary B that plays Attack Game 11.1 with respect to E: First, B chooses ! 2 {1, . . . , Q} at random.
Then, B plays the role of challenger to A: it obtains a public key pk from its own challenger, and forwards this to A; when A makes a query (mi0 , mi1 ), B computes its response ci as follows: if i > ! then c R E(pk , mi0 ) else if i = ! then B submits (mi0 , mi1 ) to its own challenger ci is set to the challenger’s response else // i < ! ci R E(pk , mi1 ). Finally, B outputs whatever A outputs. The crucial di↵erence between the proof of this theorem and that of Theorem 5.1 is that for i 6= !, adversary B can encrypt the relevant message using the public key. For b = 0, 1, let Wb be the event that B outputs 1 in Experiment 0 of its attack game. It is clear that for j = 1, . . . , Q, Pr[W0  ! = j] = pj
1
and
Pr[W1  ! = j] = pj ,
and the theorem follows by the usual telescoping sum calculation. 2 One can also consider multikey CPA security, where the adversary sees many encryptions under many public keys. In the publickey setting, semantic security implies not only CPA security, but multikey CPA security — see Exercise 11.9. 420
11.4
Encryption based on a trapdoor function scheme
In this section, we show how to use a trapdoor function scheme (see Section 10.2) to build a semantically secure publickey encryption scheme. In fact, this scheme makes use of a hash function, and our proof of security works only when we model the hash function as a random oracle (see Section 8.10.2). We then present a concrete instantiation of this scheme, based on RSA (see Section 10.3). Our encryption scheme is called ETDF , and is built out of several components: • a trapdoor function scheme T = (G, F, I), defined over (X , Y), • a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), • a hash function H : X ! K. The message space for ETDF is M, and the ciphertext space is Y ⇥ C. We now describe the key generation, encryption, and decryption algorithms for ETDF . • The key generation algorithm for ETDF is the key generation algorithm for T . • For a given public key pk , and a given message m 2 M, the encryption algorithm runs as follows: E(pk , m) :=
F (pk , x), x R X, y output (y, c).
k
H(x),
c
R
Es (k, m)
• For a given secret key sk , and a given ciphertext (y, c) 2 Y ⇥ C, the decryption algorithm runs as follows: D(sk , (y, c) ) :=
x I(sk , y), output m.
k
H(x),
m
Ds (k, c)
Thus, ETDF = (G, E, D), and is defined over (M, Y ⇥ C). The correctness property for T immediately implies the correctness property for ETDF . If H is modeled as a random oracle (see Section 8.10), one can prove that ETDF is semantically secure, assuming that T is oneway, and that Es is semantically secure. Recall that in the random oracle model, the function H is modeled as a random function O chosen at random from the set of all functions Funs[X , K]. More precisely, in the random oracle version of Attack Game 11.1, the challenger chooses O at random. In any computation where the challenger would normally evaluate H, it evaluates O instead. In addition, the adversary is allowed to ask the challenger for the value of the function O at any point of its choosing. The adversary may make any number of such “random oracle queries” at any time of its choosing. We use SSro adv[A, ETDF ] to denote A’s advantage against ETDF in the random oracle version of Attack Game 11.1. Theorem 11.2. Assume H : X ! K is modeled as a random oracle. If T is oneway and Es is semantically secure, then ETDF is semantically secure. In particular, for every SS adversary A that attacks ETDF as in the random oracle version of Attack Game 11.1, there exist an inverting adversary Bow that attacks T as in Attack Game 10.2,
421
and an SS adversary Bs that attacks Es as in Attack Game 2.1, where Bow and Bs are elementary wrappers around A, such that SSro adv[A, ETDF ] 2 · OWadv[Bow , T ] + SSadv[Bs , Es ].
(11.2)
Proof idea. Suppose the adversary sees the ciphertext (y, c), where y = F (pk , x). If H is modeled as a random oracle, then intuitively, the only way the adversary can learn anything at all about the symmetric key k used to generate c is to explicitly evaluate the random oracle representing H at the point x; however, if he could so this, we could easily convert the adversary into an adversary that inverts the function F (pk , ·), contradicting the onewayness assumption. Therefore, from the adversary’s point of view, k is completely random, and semantic security for ETDF follows directly from the semantic security of Es . In the detailed proof, we implement the random oracle using the same “faithful gnome” technique as was used to efficiently implement random functions (see Section 4.4.2); that is, we represent the random oracle as a table of input/output pairs corresponding to points at which the adversary actually queried the random oracle (as well as the point at which the challenger queries the random oracle when it runs the encryption algorithm). We also use many of the same proof techniques introduced in Chapter 4, specifically, the “forgetful gnome” technique (introduced in the proof of Theorem 4.6) and the Di↵erence Lemma (Theorem 4.7). 2 Proof. It is convenient to prove the theorem using the bitguessing versions of the semantic security game. We prove: (11.3) SSro adv⇤ [A, ETDF ] OWadv[Bow , T ] + SSadv⇤ [Bs , Es ]. Then (11.2) follows by (11.1) and (2.12). Define Game 0 to be the game played between A and the challenger in the bitguessing version of Attack Game 11.1 with respect to ETDF . We then modify the challenger to obtain Game 1. In each game, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, 1, we define Wj to be the event that ˆb = b in Game j. We will show that Pr[W1 ] Pr[W0 ] is negligible, and that Pr[W1 ] is negligibly close to 1/2. From this, it follows that SSro adv⇤ [A, ETDF ] = Pr[W0 ]
1/2
(11.4)
is also negligible. Game 0. Note that the challenger in Game 0 also has to respond to the adversary’s random oracle queries. The adversary can make any number of random oracle queries, but at most one encryption query. Recall that in addition to direct access the random oracle via explicit random oracle queries, the adversary also has indirect access to the random oracle via the encryption query, where the challenger also makes use of the random oracle. In describing this game, we directly implement the random oracle as a “faithful gnome.” This is done using an associative array Map : X ! K. The details are in Fig. 11.2. In the initialization step, the challenger prepares some quantities that will be used later in processing the encryption query. In particular, in addition to computing F (pk , x), k R K. It also sets Map[x] k, (pk , sk ) R G(), the challenger precomputes x R X , y which means that the value of the random oracle at x is equal to k. Game 1. This game is precisely the same as Game 0, except that we make our gnome “forgetful” by deleting line (3) in Fig. 11.2. Let Z be the event that the adversary queries the random oracle at the point x in Game 1. Clearly, Games 0 and 1 proceed identically unless Z occurs, and so by the Di↵erence Lemma, we 422
initialization: F (pk , x) (pk , sk ) R G(), x R X , y initialize an empty associative array Map : X ! K (2) k R K, b R {0, 1} (3) Map[x] k send the public key pk to A; (1)
upon receiving an encryption query (m0 , m1 ) 2 M2 : (4) c Es (k, mb ) send (y, c) to A; upon receiving a random oracle query x ˆ 2 X: if x ˆ2 / Domain(Map) then Map[ˆ x] R K send Map[ˆ x] to A
Figure 11.2: Game 0 challenger have Pr[W1 ]
Pr[W0 ] Pr[Z].
(11.5)
If event Z happens, then one of the adversary’s random oracle queries is the inverse of y under F (pk , ·). Moreover, in Game 1, the value x is used only to define y = F (pk , x), and nowhere else. Thus, we can use adversary A to build an efficient adversary Bow that breaks the onewayness assumption for T with an advantage equal to Pr[Z]. Here is how adversary Bow works in detail. This adversary plays Attack Game 10.2 against a challenger Cow , and plays the role of challenger to A as in Fig. 11.2, except with the following lines modified as indicated: (1) (3)
obtain (pk , y) from Cow (deleted)
Additionally, when A terminates: if F (pk , x ˆ) = y for some x ˆ 2 Domain(Map) then output x ˆ else output “failure”. To analyze Bow , we may naturally view Game 1 and the game played between Bow and Cow as operating on the same underlying probability space. By definition, Z occurs if and only if x 2 Domain(Map) when Bow finishes its game. Therefore, Pr[Z] = OWadv[Bow , T ].
(11.6)
Observe that in Game 1, the key k is only used to encrypt the challenge plaintext. As such, the adversary is essentially attacking Es as in the bitguessing version of Attack Game 2.1 at this 423
point. More precisely, we derive an efficient SS adversary Bs based on Game 1 that uses A as a subroutine, such that (11.7) Pr[W1 ] 1/2 = SSadv⇤ [Bs , Es ]. Adversary Bs plays the bitguessing version of Attack Game 2.1 against a challenger Cs , and plays the role of challenger to A as in Fig. 11.2, except with the following lines modified as indicated: (2) (3) (4)
(deleted) (deleted) forward (m0 , m1 ) to Cs , obtaining c
Additionally, when A outputs ˆb: output ˆb To analyze Bs , we may naturally view Game 1 and the game played between Bs and Cs as operating on the same underlying probability space. By construction, Bs and A output the same thing, and so (11.7) holds. Combining (11.4), (11.5), (11.6), and (11.7), yields (11.3). 2
11.4.1
Instantiating ETDF with RSA
Suppose we now use RSA (see Section 10.3) to instantiate T in the above encryption scheme ETDF . This scheme is parameterized by two quantities: the length ` of the prime factors of the RSA modulus, and the encryption exponent e, which is an odd, positive integer. Recall that the RSA scheme does not quite fit the definition of a trapdoor permutation scheme, because the domain of the trapdoor permutation is not a fixed set, but varies with the public key. Let us assume that X is a fixed set into which we may embed Zn , for every RSA modulus n generated by RSAGen(`, e) (for example, we could take X = {0, 1}2` ). The scheme also makes use of a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), as well as a hash function H : X ! K. The basic RSA encryption scheme is ERSA = (G, E, D), with message space M and ciphertext space X ⇥ C, where • the key generation algorithm runs as follows: G() :=
(n, d) R RSAGen(`, e), output (pk , sk );
pk
(n, e),
sk
(n, d)
• for a given public key pk = (n, e), and message m 2 M, the encryption algorithm runs as follows: E(pk , m) :=
xe , k x R Zn , y output (y, c) 2 X ⇥ C;
H(x),
c
R
Es (k, m)
• for a given secret key sk = (n, d), and a given ciphertext (y, c) 2 X ⇥ C, where y represents an element of Zn , the decryption algorithm runs as follows: D(sk , (y, c) ) :=
x yd , k output m.
H(x),
424
m
Ds (k, c)
Theorem 11.3. Assume H : X ! K is modeled as a random oracle. If the RSA assumption holds for parameters (`, e), and Es is semantically secure, then ERSA is semantically secure. In particular, for any SS adversary A that attacks ERSA as in the random oracle version of Attack Game 11.1, there exist an RSA adversary Brsa that breaks the RSA assumption for (`, e) as in Attack Game 10.3, and an SS adversary Bs that attacks Es as in Attack Game 2.1, where Brsa and Bs are elementary wrappers around A, such that SSro adv⇤ [A, ERSA ] RSAadv[Brsa , `, e] + SSadv⇤ [Bs , Es ].
Proof. The proof of Theorem 11.2 carries over, essentially unchanged. 2
11.5
ElGamal encryption
In this section we show how to build a publickey encryption scheme from DiffieHellman. Security will be based on either the CDH or DDH assumptions from Section 10.5. The encryption scheme is a variant of a scheme first proposed by ElGamal, and we call it EEG . It is built out of several components: • a cyclic group G of prime order q with generator g 2 G, • a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), • a hash function H : G ! K. The message space for EEG is M, and the ciphertext space is G ⇥ C. We now describe the key generation, encryption, and decryption algorithms for EEG . • the key generation algorithm runs as follows: G() :=
↵ R Zq , u g↵ pk u, sk ↵ output (pk , sk );
• for a given public key pk = u 2 G and message m 2 M, the encryption algorithm runs as follows: E(pk , m) :=
Zq , v output (v, c); R
g ,
w
u ,
k
H(w),
c
Es (k, m)
• for a given secret key sk = ↵ 2 Zq and a ciphertext (v, c) 2 G ⇥ C, the decryption algorithm runs as follows: D(sk , (v, c) ) :=
w v↵ , k output m.
H(w),
m
Ds (k, c)
Thus, EEG = (G, E, D), and is defined over (M, G ⇥ C). Note that the description of the group G and generator g 2 G is considered to be a system parameter, rather than part of the public key.
425
11.5.1
Semantic security of ElGamal in the random oracle model
We shall analyze the security of EEG under two di↵erent sets of assumptions. In this section we do the analysis modeling H : G ! K as a random oracle, under the CDH assumption for G, and the assumption that Es is semantically secure. In the next section we analyze EEG without the random oracle model, but using the stronger DDH assumption for G. Theorem 11.4. Assume H : G ! K is modeled as a random oracle. If the CDH assumption holds for G, and Es is semantically secure, then EEG is semantically secure. In particular, for every SS adversary A that plays the random oracle version of Attack Game 11.1 with respect to EEG , and makes at most Q queries to the random oracle, there exist a CDH adversary Bcdh that plays Attack Game 10.5 with respect to G, and an SS adversary Bs that plays Attack Game 2.1 with respect to Es , where Bcdh and Bs are elementary wrappers around A, such that SSro adv[A, EEG ] 2Q · CDHadv[Bcdh , G] + SSadv[Bs , Es ]. (11.8)
Proof idea. Suppose the adversary sees the ciphertext (v, c), where v = g . If H is modeled as a random oracle, then intuitively, the only way the adversary can learn anything at all about the symmetric key k used to generate c is to explicitly evaluate the random oracle representing H at the point w = v ↵ ; however, if he could so this, we could convert the adversary into an adversary that breaks the CDH assumption for G. One wrinkle is that we cannot recognize the correct solution to the CDH problem when we see it (if the DDH assumption is true), so we simply guess by choosing at random from among all of the adversary’s random oracle queries. This is where the factor of Q in (11.8) comes from. So unless the adversary can break the CDH assumption, from the adversary’s point of view, k is completely random, and semantic security for EEG follows directly from the semantic security of Es . 2 Proof. It is convenient to prove the theorem using the bitguessing version of the semantic security game. We prove: SSro adv⇤ [A, EEG ] Q · CDHadv[Bcdh , G] + SSadv⇤ [Bs , Es ].
(11.9)
Then (11.8) follows from (11.1) and (2.12). We define Game 0 to be the game played between A and the challenger in the bitguessing version of Attack Game 11.1 with respect to EEG . We then modify the challenger to obtain Game 1. In each game, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, 1, we define Wj to be the event that ˆb = b in Game j. We will show that Pr[W1 ] Pr[W0 ] is negligible, and that Pr[W1 ] is negligibly close to 1/2. From this, it follows that SSro adv⇤ [A, EEG ] = Pr[W0 ]
1/2
(11.10)
is negligible. Game 0. The adversary can make any number of random oracle queries, but at most one encryption query. Again, recall that in addition to direct access the random oracle via explicit random oracle queries, the adversary also has indirect access to the random oracle via the encryption query, where the challenger also makes use of the random oracle. The random oracle is implemented using an associative array Map : G ! K. The details are in Fig. 11.3. At line (3), we e↵ectively set the random oracle at the point w to k. 426
initialization: g↵, v g ,w g↵ ↵, R Zq , u initialize an empty associative array Map : G ! K (2) k R K, b R {0, 1} (3) Map[w] k send the public key u to A; (1)
upon receiving an encryption query (m0 , m1 ) 2 M2 : (4) c Es (k, mb ) send (v, c) to A; upon receiving a random oracle query w ˆ 2 G: if w ˆ2 / Domain(Map) then Map[w] ˆ R K send Map[w] ˆ to A
Figure 11.3: Game 0 challenger Game 1. This is the same as Game 0, except we delete line (3) in Fig. 11.3. Let Z be the event that the adversary queries the random oracle at w in Game 1. Clearly, Games 0 and 1 proceed identically unless Z occurs, and so by the Di↵erence Lemma, we have Pr[W1 ]
Pr[W0 ] Pr[Z].
(11.11)
If event Z happens, then one of the adversary’s random oracle queries is the solution w to the instance (u, v) of the CDH problem. Moreover, in Game 1, the values ↵ and are only needed to compute u and v, and nowhere else. Thus, we can use adversary A to build an adversary Bcdh to break the CDH assumption: we simply choose one of the adversary’s random oracle queries at random, and output it — with probability at least Pr[Z]/Q, this will be the solution to the given instance of the CDH problem. In more detail, adversary Bcdh plays Attack Game 10.5 against a challenger Ccdh , and plays the role of challenger to A as in Fig. 11.3, except with the following lines modified as indicated: (1) (3)
obtain (u, v) from Ccdh (deleted)
Additionally, when A terminates: if Domain(Map) 6= ; then w ˆ R Domain(Map), output w ˆ else output “failure” To analyze Bcdh , we may naturally view Game 1 and the game played between Bcdh and Ccdh as operating on the same underlying probability space. By definition, Z occurs if and only if w 2 Domain(Map) when Bcdh finishes its game. Moreover, since Domain(Map) Q, it follows that CDHadv[Bcdh , G] Pr[Z]/Q. (11.12) 427
Observe that in Game 1, the key k is only used to encrypt the challenge plaintext. We leave it to the reader to describe an efficient SS adversary Bs that uses A as a subroutine, such that Pr[W1 ]
1/2 = SSadv⇤ [Bs , Es ].
(11.13)
Combining (11.10), (11.11), (11.12), and (11.13), yields (11.9), which completes the proof of the theorem. 2
11.5.2
Semantic security of ElGamal without random oracles
As we commented in Section 8.10.2, security results in the random oracle model do not necessarily imply security in the real world. When it does not hurt efficiency, it is better to avoid the random oracle model. By replacing the CDH assumption by the stronger, but still reasonable, DDH assumption, and by making an appropriate, but reasonable, assumption about H, we can prove that the same system EEG is semantically secure without resorting to the random oracle model. We thus obtain two security analyses of EEG : one in the random oracle model, but using the CDH assumption. The other, without the random oracle model, but using the stronger DDH assumption. We are thus using the random oracle model as a hedge: in case the DDH assumption turns out to be false in the group G, the scheme remains secure assuming CDH holds in G, but in a weaker random oracle semantic security model. In Exercise 11.13 we develop yet another analysis of ElGamal without random oracles, but using a weaker assumption than DDH called hash DiffieHellman (HDH) which more accurately captures the exact requirement to needed to prove security. To carry out the analysis using the DDH assumption in G we make a specific assumption about the hash function H : G ! K, namely that H is a secure key derivation function, or KDF for short. We already introduced a very general notion of a key derivation function in Section 8.10. What we describe here is more focused and tailored precisely to our current situation. Intuitively, H : G ! K is a secure KDF if no efficient adversary can e↵ectively distinguish between H(w) and k, where w is randomly chosen from G, and k is randomly chosen from K. To be somewhat more general, we consider an arbitrary, efficiently computable hash function F : X ! Y, where X and Y are arbitrary, finite sets. Attack Game 11.3 (secure key derivation). For a given hash function F : X ! Y, and for a given adversary A, we define two experiments. Experiment b
(b = 0, 1):
• The challenger computes
x
R
X,
y0
F (x),
y1
R
Y,
and sends yb to the adversary. • The adversary outputs a bit ˆb 2 {0, 1}. If Wb is the event that A outputs 1 in Experiment b, then we define A’s advantage with respect to F as KDFadv[A, F ] := Pr[W0 ] Pr[W1 ] . 2
428
Definition 11.5 (secure key derivation). A hash function F : X ! Y is a secure KDF if for every efficient adversary A, the value KDFadv[A, F ] is negligible. It is plausible to conjecture that an “o↵ the shelf” hash function, like SHA256 or HKDF (see Section 8.10.5), is a secure KDF. In fact, one may justify this assumption modeling the hash function as a random oracle; however, using this explicit computational assumption, rather than the random oracle model, yields more meaningful results. One may even build a secure KDF without making any assumptions at all: the construction in Section 8.10.4 based on a universal hash function and the leftover hash lemma yields an unconditionally secure KDF. Even though this construction is theoretically attractive and quite efficient, it may not be a wise choice from a security point of view: as already discussed above, if the DDH turns out to be false, we can still rely on the CDH in the random oracle model, but for that, it is better to use something based on SHA256 or HKDF, which can more plausibly be modeled as a random oracle. Theorem 11.5. If the DDH assumption holds for G, H : G ! K is a secure KDF, and Es is semantically secure, then EEG is semantically secure. In particular, for every SS adversary A that plays Attack Game 11.1 with respect to EEG , there exist a DDH adversary Bddh that plays Attack Game 10.6 with respect to G, a KDF adversary Bkdf that plays Attack Game 11.3 with respect to H, and an SS adversary Bs that plays Attack Game 2.1 with respect to Es , where Bddh , Bkdf , and Bs are elementary wrappers around A, such that (11.14) SSadv[A, EEG ] 2 · DDHadv[Bddh , G] + 2 · KDFadv[Bkdf , H] + SSadv[Bs , Es ].
Proof idea. Suppose the adversary sees the ciphertext (v, c), where v = g and c is a symmetric encryption created using the key k := H(u ). Suppose the challenger replaces w = u by a random independent group element w ˜ 2 G and constructs k as k := H(w). ˜ By the DDH assumption the ˜ and hence its advantage is only negligibly adversary cannot tell the di↵erence between u and w changed. Under the KDF assumption, k := H(w) ˜ looks like a random key in K, independent of the adversary’s view, and therefore security follows by semantic security of Es . 2 Proof. More precisely, it is convenient to prove the theorem using the bitguessing version of the semantic security game. We prove: SSadv⇤ [A, EEG ] DDHadv[Bddh , G] + KDFadv[Bkdf , H] + SSadv⇤ [Bs , Es ].
(11.15)
Then (11.14) follows by (11.1) and (2.12). Define Game 0 to be the game played between A and the challenger in the bitguessing version of Attack Game 11.1 with respect to EEG . We then modify the challenger to obtain Games 1 and 2. In each game, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, 1, 2, we define Wj to be the event that ˆb = b in Game j. We will show that Pr[W2 ] Pr[W0 ] is negligible, and that Pr[W2 ] is negligibly close to 1/2. From this, it follows that SSadv⇤ [A, EEG ] = Pr[W0 ]
1/2
is negligible. Game 0. The logic of the challenger in this game is presented in Fig. 11.4. 429
(11.16)
(1) (2)
initialization: ↵ ,u g↵, v ↵, R Zq , k H(w) b R {0, 1} send the public key u to A;
g ,w
g
upon receiving (m0 , m1 ) 2 M2 : c Es (k, mb ), send (v, c) to A
Figure 11.4: Game 0 challenger Game 1. We first play our “DDH card.” The challenger in this game is as in Fig. 11.4, except that line (1) is modified as follows: (1)
R
↵,
Zq ,
R
Zq , u
g↵, v
g ,w
g
We describe an efficient DDH adversary Bddh that uses A as a subroutine, such that Pr[W0 ]
Pr[W1 ] = DDHadv[Bddh , G].
(11.17)
Adversary Bddh plays Attack Game 10.6 against a challenger Cddh , and plays the role of challenger to A as in Fig. 11.4, except with line (1) modified as follows: (1)
obtain (u, v, w) from Cddh
Additionally, when A outputs ˆb: if b = ˆb then output 1 else output 0 Let p0 be the probability that Bddh outputs 1 when Cddh is running Experiment 0 of the DDH Attack Game 10.6, and let p1 be the probability that Bddh outputs 1 when Cddh is running Experiment 1. By definition, DDHadv[Bddh , G] = p1 p0 . Moreover, if Cddh is running Experiment 0, then adversary A is playing our Game 0, and so p0 = Pr[W0 ], and if Cddh is running Experiment 1, then A is playing our Game 1, and so p1 = Pr[W1 ]. Equation (11.17) now follows immediately. Game 2. Observe that in Game 1, w is completely random, and is used only as an input to H. This allows us to play our “KDF card.” The challenger in this game is as in Fig. 11.4, except with the following lines modified as indicated: (1) (2)
↵, k
R R
Zq ,
R
Zq , u
g↵, v
g ,w
g
K
We may easily derive an efficient KDF adversary Bkdf that uses A as a subroutine, such that Pr[W1 ]
Pr[W2 ] = KDFadv[Bkdf , H].
(11.18)
Adversary Bkdf plays Attack Game 11.3 against a challenger Ckdf , and plays the role of challenger to A as in Fig. 11.4, except with the following lines modified as indicated: 430
↵, R Zq , u g↵, v obtain k from Ckdf
(1) (2)
g ,
R
Zq , w
g
Additionally, when A outputs ˆb: if b = ˆb then output 1 else output 0 We leave it to the reader to verify (11.18). Observe that in Game 2, the key k is only used to encrypt the challenge plaintext. As such, the adversary is essentially just playing the SS game with respect to Es at this point. We leave it to the reader to describe an efficient SS adversary Bs that uses A as a subroutine, such that Pr[W2 ]
1/2 = SSadv⇤ [Bs , Es ].
(11.19)
Combining (11.16), (11.17), (11.18), and (11.19), yields (11.15), which completes the proof of the theorem. 2
11.6
Threshold decryption
We next discuss an important technique used to protect the secret key sk in a public key encryption scheme. Suppose sk is stored on a server, and that server is used to decrypt incoming ciphertexts. If the server is compromised, and the key is stolen, then all ciphertexts ever encrypted under the corresponding publickey can be decrypted by the attacker. For this reason, important secret keys are sometimes stored in a special hardware component, called a hardware security module (HSM) that responds to decryption requests, but never exports the secret key in the clear. An attacker who compromises the server can temporarily use the key, but cannot steal the key and use it o✏ine. Another approach to protecting a secret key is to split it into a number of pieces, called shares, and require that all the shares must be present in order to decrypt a ciphertext. Each share can be stored on a di↵erent machine so that all the machines must cooperate in order to decrypt a ciphertext. Decryption fails if even one machine does not participate. Consequently, to steal the secret key, an attacker must break the security of all the machines, and this can be harder than compromising a single machine. In what follows, we use s to denote the total number of shares. While splitting the key makes it harder to steal, it also hurts availability. If even a single share is lost, decryption becomes impossible. For this reason we often require that decryption can proceed even if only t of the s shares are available, for some 0 < t s. For security, t 1 shares should reveal nothing about the key sk , and should not help the adversary decrypt ciphertexts. Typical values for t and s are 3outof5 or 5outof8; however some applications require larger values for t and s. In a 3outof5 sharing, stealing only two shares should reveal nothing helpful to the adversary. Threshold decryption. Ideally, during decryption, the secret key sk is never reconstituted in a single location. This ensures that there is no single point of failure that an adversary can attack to steal the key. In such a system, there are s key servers, and an additional entity called a combiner that orchestrates the decryption process. The combiner takes as input a ciphertext c to decrypt, and forwards c to all the key servers. Every online server applies its key share to c, and 431
c
c
c
c
c
sk 0
sk 1
sk 2
sk 3
sk 4
c˜2
c˜0 c
key servers
c˜4
combiner c
m
The combiner sends the given ciphertext c to all five key servers. Three servers respond, enabling the combiner to construct and output the plaintext message m. Figure 11.5: Threshold decryption using three responses from five key servers. sends back a “partial decryption.” Once t responses are received from the key servers, the combiner can construct the complete decryption of c. The entire process is shown in Fig. 11.5. Overall, the system should decrypt c without reconstituting the key sk in a single location. Such a system is said to support threshold decryption. Definition 11.6. A publickey threshold decryption scheme E = (G, E, D, C) is a tuple of four efficient algorithms: • G is a probabilistic algorithm that is invoked as (pk , sk 1 , . . . , sk s ) R G(s, t) to generate a toutofs shared key. It outputs a public key pk and s shares SK := {sk 1 , . . . , sk s } of the decryption key. • E is an encryption algorithm as in a public key encryption scheme, invoked as c
R
E(pk , m).
D(sk i , c), where sk i is one of the key • D is a deterministic algorithm that is invoked as c0 0 shares output by G, c is a ciphertext, and c is a partial decryption of c using sk i . • C is a deterministic algorithm that is invoked as m C(c, c01 , . . . , c0t ), where c is a ciphertext, and c01 , . . . , c0t are some t partial decryptions of c, computed using t distinct key shares. • As usual, decryption should correctly decrypt wellformed ciphertexts; specifically, for all possible outputs (pk , sk 1 , . . . , sk s ) of G(s, t), all messages m, and all tsize subsets {sk 01 , . . . , sk 0t } of sk , for all outputs c of E(pk , m), we have C( c, D(sk 01 , c), . . . , D(sk 0t , c) ) = m. A publickey threshold decryption scheme is secure if an adversary that completely compromises t 1 of the key servers, and can eavesdrop on the output of the remaining key servers, cannot break semantic security. We will define security more precisely after we look at some constructions. Note that Definition 11.6 requires that t and s be specified at key generation time. However, all the schemes in this section can be extended so that both t and s can be changed after the secret key shares are generated, without changing the public key pk . Combinatorial threshold decryption. Recall that in Exercise 2.21 we saw how a symmetric decryption key k can be split into three shares, so that any two shares can be used to decrypt a 432
given ciphertext, but a single share cannot. The scheme can be generalized so that k can be split into s shares and any t s can be used to decrypt, but t 1 shares cannot. The communication pattern during decryption is a little di↵erent than the one shown in Fig. 11.5, but nevertheless, the system satisfies our goal of decrypting without ever reconstituting the key k in a single location. The difficulty with the scheme in Exercise 2.21 is that its performance degrades rapidly as t and s grow. Even supporting a small number of shares, say a 5outof8 sharing, requires a ciphertext that is over fourteen times as long as a nonthreshold ciphertext. ElGamal threshold decryption. As we will shortly see, the ElGamal encryption scheme (Section 11.5) supports a very efficient threshold decryption mechanism, even for large t and s. In Exercise 11.16 we look at RSA threshold decryption.
11.6.1
Shamir’s secret sharing scheme
Our threshold version of ElGamal encryption is based on a technique, which has numerous other application, called secret sharing. Suppose Alice has a secret ↵ 2 Z, where Z is some finite set. She wishes to generate s shares of ↵, each belonging to some finite set Z 0 , and denoted ↵1 , . . . , ↵s 2 Z 0 , so that the following property is satisfied: any t of the s shares are sufficient to reconstruct ↵, but every set of t 1 shares reveals nothing about ↵. This sharing lets Alice give one share to each of her s friends, so that any t friends can help her recover ↵, but t 1 friends learn nothing. Such a scheme is called a secret sharing scheme. Definition 11.7. A secret sharing scheme over Z is a pair of efficient algorithms (G, C): • G is a probabilistic algorithm that is invoked as (↵1 , . . . , ↵s ) R G(s, t, ↵), where 0 < t s and ↵ 2 Z, to generate a toutofs sharing of ↵. It outputs s shares SK := {↵1 , . . . , ↵s }. • C is a deterministic algorithm that is invoked as ↵
C(↵10 , . . . , ↵t0 ), to recover ↵.
• Correctness: we require that for every ↵ 2 Z, every set of s shares SK output by G(s, t, ↵), and every tsize subset {↵10 , . . . , ↵t0 } of SK, we have that C(↵10 , . . . , ↵t0 ) = ↵. Intuitively, a secret sharing scheme is secure if every set of t 1 shares output by G(s, t, ↵) reveals nothing about ↵. To define this notion formally, it will be convenient to use the following notation: for a set S ✓ {1, . . . , s}, we denote by G(s, t, ↵)[S] the set of shares output by G at positions indicated by S. For example, G(s, t, ↵)[{1, 3, 4}] is the set {↵1 , ↵3 , ↵4 }. Definition 11.8. A secret sharing scheme (G, C) over Z is secure if for every ↵, ↵0 2 Z, and every subset S of {1, . . . , s} of size t 1, the distribution G(s, t, ↵)[S] is identical to the distribution G(s, t, ↵0 )[S]. The definition implies that by looking at t for all ↵ and ↵0 in Z. Hence, looking at only t
1 shares, one cannot tell if the secret is ↵ or ↵0 , 1 shares reveals nothing about the secret.
Shamir secret sharing. An elegant secret sharing scheme over Zq , where q is prime, is due to Shamir. This scheme makes use of the following general fact about polynomial interpolation: a polynomial of degree at most t 1 is completely determined by t points on the polynomial. For example, two points determine a line, and three points determine a parabola. This general fact not 433
only holds for the real numbers and complex numbers, but over any algebraic domain in which all nonzero elements have a multiplicative inverse. Such a domain is called a field. When q is prime, Zq is a field, and so this general fact holds here as well. Shamir’s scheme (Gsh , Csh ) is a toutofs secret sharing scheme over Zq that requires that q > s, and works as follows: • Gsh (s, t, ↵): choose random a1 , . . . , at f (x) := at
1x
t 1
Notice that f has degree at most t
1
+ at
R
Zq and define the polynomial
2x
t 2
+ . . . + a1 x + ↵ 2 Zq [x].
1 and that f (0) = ↵.
Next, choose arbitrary s nonzero points x1 , . . . , xs in Zq (for example, we could just use the points 1, . . . , s in Zq ). f (xi ) 2 Zq , and define ↵i := (xi , yi ). For i = 1, . . . , s compute yi Output the s shares ↵1 , . . . , ↵s 2 Z2q . • Csh (↵10 , . . . , ↵t0 ): an input of t valid shares corresponds to t points on the polynomial f , and these t points completely determine f . Algorithm Csh interpolates the polynomial f and outputs ↵ := f (0). The description of algorithm Csh needs a bit more explanation. A simple method for interpolating the polynomial of degree at most t 1 from t points is called Lagrange interpolation. Let us see how it works. Given t shares ↵i0 = (x0i , yi0 ) for i = 1, . . . , t, define t polynomials: Li (x) :=
t Y x x0i
j=1 j6=i
x0j x0j
2 Zq [x]
for i = 1, . . . , t.
It is not difficult to verify that: Li (x0i ) = 1 and Li (x0j ) = 0 for all j 6= i in {1, . . . , t}. Next, consider the polynomial g(x) := L1 (x) · y10 + . . . + Lt (x) · yt0 2 Zq [x]
Again, it is not difficult to see that g(x0i ) = yi0 = f (x0i ) for all i = 1, . . . , t. Since both f and g are polynomials of degree t 1, and they match at t points, they must be the same polynomial (here is we use our general fact about polynomial interpolation). Therefore, ↵ = f (0) = g(0), and in particular t t X Y x0j 0 ↵ = g(0) = where 2 Zq . (11.20) i · yi i := Li (0) = x0i x0j i=1
j=1 j6=i
The scalars 1 , . . . , t 2 Zq are called Lagrange coefficients. Using (11.20) we can now describe algorithm Csh in more detail. Given a set of t 1 shares, the algorithm first computes the Lagrange coefficients 1 , . . . , t 2 Zq . Computing these quantities requires division, but since q is prime, this is always well defined. It then computes ↵ using the linear combination on the left side of (11.20). Note that the Lagrange coefficients 1 , . . . , t do not depend on the secret ↵, and can be precomputed if one knows ahead of time which shares will be used to reconstruct ↵. 434
Security.
It remains to show that this secret sharing scheme is secure, as in Definition 11.8.
Theorem 11.6. Shamir’s secret sharing scheme (Gsh , Csh ) is secure. Proof. To prove the theorem, we shall show that for every ↵ 2 Zq , any set of t 1 shares (x01 , y10 ), . . . , (x0t 1 , yt0 1 ) has the property that the ycoordinates y10 , . . . , yt0 1 are uniformly and independently distributed over Zq . So let ↵ and x01 , . . . , x0t 1 be fixed. Claim. Consider the map that sends (a1 , . . . , at 1 ) 2 Ztq 1 (as chosen by Gsh (s, t, ↵)) to 0 (y1 , . . . , yt0 1 ) 2 Ztq 1 , which are the ycoordinates of the shares whose xcoordinates are x01 , . . . , x0t 1 . Then this map is onetoone. The theorem follows from the claim, since if (a1 , . . . , at 1 ) is chosen uniformly over Ztq 1 , then (y10 , . . . , yt0 1 ) must also be uniformly distributed over Ztq 1 . Finally, to prove the claim, suppose by way of contradiction that this map is not onetoone. This would imply the existence of two distinct polynomials g(x), h(x) 2 Zq [x] of degree at most t 2, such that the polynomials ↵ + xg(x) and ↵ + xh(x) agree at the t 1 nonzero points x01 , . . . , x0t 1 . But then this implies that g(x) and h(x) themselves agree at these same t 1 points, which contradicts our basic fact about polynomial interpolation. 2
11.6.2
ElGamal threshold decryption
For any publickey encryption scheme, one can use Shamir secret sharing to share the secret decryption key sk , in a toutofs fashion, among s servers. Then any t servers can help the combiner reconstruct the secret key and decrypt a given ciphertext. However, this creates a single point of failure: an adversary who compromises the combiner during decryption will learn sk in the clear. In this section we show how to enhance ElGamal decryption, so that decryption can be done with the help of t servers, as in Fig. 11.5, but without reconstituting the key at a single location. We first describe the scheme, and then define and prove security. ElGamal threshold decryption. Recall that the ElGamal encryption scheme (Section 11.5) uses a group G of prime order q with generator g 2 G, a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), and a hash function H : G ! K. The secret key sk is an element ↵ 2 Zq , and a ciphertext (v, c) 2 G ⇥ C is decrypted by first computing w v↵. To support toutofs threshold decryption, the key generation algorithm first generates a toutofs Shamir secret sharing of the ElGamal decryption key ↵ 2 Zq . The resulting shares, (xi , yi ) for i = 1, . . . , s, are the shares of the decryption key ↵, and each key server is given one share. Now, to decrypt an ElGamal ciphertext (v, c), it suffices for some t key servers to send the partial decryption (xi , v yi ) 2 Zq ⇥ G to the combiner. Once the combiner receives t partial decryptions c0i = (xi , v yi ) for i = 1, . . . , t, it decrypts the ciphertext as follows: First, the combiner uses x1 , . . . , xt to compute the Lagrange coefficients 1 , . . . , t 2 Zq as in Eq. (11.20). Next, it computes w (v y1 ) 1 · (v y2 ) 2 · · · (v yt ) t 2 G. By (11.20) we know that w = v (y1 ·
1 +···+yt · t )
= v↵.
(11.21)
This w = v ↵ is sufficient to decrypt the ciphertext (v, c), as in normal ElGamal decryption. Observe that during decryption, the ElGamal decryption key ↵ was never assembled in a single location. The complete ElGamal threshold decryption system EthEG = (G, E, D, C) works as follows: 435
• Key generation runs as follows, using Shamir’s secret sharing scheme (Gsh , Csh ): G(s, t) :=
pk := u g↵ ↵ R Zq , R (x1 , y1 ), . . . , (xs , ys ) Gsh (s, t, ↵) := for i = 1, . . . , s set sk i (xi , yi ) output (pk , sk 1 , . . . , sk s )
• The encryption algorithm E(pk , m) is the same as in ElGamal encryption in Section 11.5. It outputs a pair (v, c) 2 G ⇥ C. • for a given secret key share sk i = (x, y) 2 Zq ⇥ G and a ciphertext (v, c) 2 G ⇥ C, the partial decryption algorithm runs as follows: D(sk i , (v, c) ) :=
w vy , output c0 := (x, w) 2 Zq ⇥ G.
• given a ciphertext (v, c) 2 G ⇥ C, and t partial decryptions c0i = (xi , wi ) for i = 1, . . . , t, the combine algorithm runs as follows: C (v, c), c01 , . . . , c0t := use x1 , . . . , xt to compute 1 , . . . , t 2 Zq as in (11.20) H(w), m Ds (k, c) (⇤) set w w1 1 · w2 2 · · · wt t 2 G, k output m The combine algorithm works correctly because, as explained in (11.21), the quantity w computed on line (⇤) satisfies w = v ↵ , which is then used to derive the symmetric encryption key k needed to decrypt c. ElGamal threshold decryption is secure. First, let us define more precisely what it means for a threshold decryption scheme to be secure. As usual, this is done by defining an attack game. Just as in Attack Game 11.1, our adversary will be allowed to make a single encryption query, in which he submits a pair of messages to the challenger, and obtains an encryption of one of them. However, to capture the notion of security we are looking for in a threshold decryption scheme, in addition to the public key, the adversary also gets to see t 1 shares of the secret key of its choice. Additionally, we want to capture the notion that the combiner cannot become a single point of failure. To this end, we allow the adversary to make any number of combiner queries: in such a query, the adversary sumbits a single message to the challenger, and gets to see not only its encryption, but also all s of the corresponding partial decryptions of the ciphertext. Our security definition, given below, allows the adversary to eavesdrop on all traffic sent to the combiner. A more powerful adversary might completely compromise the combiner, and tamper with what it sends to the key servers. We do not consider such adversaries here, but will come back to this question in Chapter 16. Attack Game 11.4 (threshold decryption semantic security). For a publickey threshold decryption scheme E = (G, E, D, C) defined over (M, C), and for a given adversary A, we define two experiments, parameterized by integers 0 < t s. Experiment b (b = 0, 1):
436
• Setup: the adversary chooses a set S ✓ {1, . . . , s} of size t 1 and gives it to the challenger. The challenger runs (pk , sk 1 , . . . , sk s ) R G(s, t) and sends pk and {sk i }i2S to the adversary. • The adversary queries the challenger several times. Each query can be one of two types: – Combiner query: for j = 1, 2, . . . , the jth such query is a message mj 2 M. The D(sk i , cj ), for challenger computes cj R E(pk , mj ) and the s partial decryptions c0j,i i = 1, . . . , s. The challenger sends cj and c0j,1 , . . . , c0j,s to the adversary. – Single encryption query: The adversary sends m0 , m1 2 M, of the same length, to the challenger. The challenger computes c R E(pk , mb ), and sends c to the adversary. The adversary may only issue a single encryption query (which may be preceded or followed by any number of combiner queries). • The adversary outputs a bit ˆb 2 {0, 1}. If Wb is the event that A outputs 1 in Experiment b, define A’s advantage with respect to E as thSSadv[A, E] := Pr[W0 ]
Pr[W1 ] .
2
Definition 11.9 (threshold decryption semantic security). A publickey threshold decryption scheme E is semantically secure if for all efficient adversaries A, the value thSSadv[A, E] is negligible. Next, we argue that the ElGamal threshold decryption scheme EthEG is semantically secure. The proof is very similar to the proof of Theorem 11.5. Theorem 11.7. If EEG is semantically secure, then EthEG is threshold decryption semantically secure. In particular, for every adversary A that attacks EthEG as in Attack Game 11.4, there exists an adversary B that attacks EEG as in Attack Game 11.1, such that thSSadv[A, EthEG ] = SSadv[B, EEG ].
Proof. We design B to play the role of challenger to A. When A receives pk = g ↵ from its own challenger, we need to have A provide to B not only pk , but also t 1 key shares. By Theorem 11.6, we know that (Gsh , Csh ) satisfies Definition 11.7, which means that we can generate the required ↵, r, s) for an arbitrary ↵ ˆ 2 Zq . In fact, by the proof of t 1 key shares by just running Gsh (ˆ of Theorem 11.6, we know that we can just generate the ycoordinates of the required shares by choosing elements of Zq uniformly and independently. When A makes its single encryption query, B forwards this query to its own challenger, and forwards the response from the challenger back to A. Whenever A outputs a bit ˆb 2 {0, 1}, our adversary B outputs the same bit ˆb. To finish the proof, we have to show how our B can faithfully respond to all of A’s combiner queries. Once we do this, the proof will be finished: B will have the same advantage in its attack game that A has in its attack game. Let (x0i , yi0 ) for i = 1, . . . , t 1 be the key shares that were given to A. Let m 2 M be a combiner R Zq and computing v g , w u , c query. Our B first encrypts m by choosing a random Es (H(w), m). Now, let (x, y) be some key share. Our B needs to compute the partial decryption c0 := (x, v y ). There are two cases: 437
• If x 2 {x01 , . . . , x0t
1}
then B knows y and can easily compute c0 := (x, v y ).
• Otherwise, our B can compute v y without knowing y, as follows. It uses (11.20) to compute the t Lagrange coefficients , 1 , . . . , t 1 2 Zq corresponding to the t points x, x01 , . . . , x0t 1 2 Zq . Although B does not know ↵ or y, it knows that ·y+
↵= By multiplying both sides by u =g
·↵
=g
· ·y
1
· y10 + . . . +
t 1
· yt0
1.
and exponentiating, it follows that ·g
(
0 0 1 ·y1 +···+ t 1 ·yt 1 )
= (v y ) · g
(
0 0 1 ·y1 +···+ t 1 ·yt 1 )
.
Since v y is the only unknown in this equation, B can easily solve for v y , and obtain the required value. In conclusion, we see that B can compute all the required partial decryptions c0 := (x, v y ), and send them to the adversary, along with the ciphertext (v, c). 2 Further enhancements. The threshold decryption scheme EthEG can be strengthened in several ways. First, the system EthEG easily generalizes to more flexible access structures than strict threshold. For example, it is easy to extend the scheme to support the following access structure: decryption is possible if key server number 1 participates, and at least t of the remaining s 1 key servers participate. We explore more general access structures in Exercise 11.15. Another enhancement, called proactive security, further strengthens the system by forcing the adversary to break into all s servers within a short period of time, say ten minutes [53]. Otherwise, the adversary gets nothing. This is done by having the key servers proactively refresh the sharing of their secret key every ten minutes, without changing the public key. Finally, key generation can be strengthened so that the secret key ↵ is not generated in a central location. Instead, the s key servers engage in a distributed computation to generate the key shares [45]. This way the secret key ↵ is always stored in shared form, from inception to final retirement.
11.7
Fun application: oblivious transfer from DDH
To be written.
11.8
Notes
Citations to the literature to be added.
11.9
Exercises
11.1 (Simple PRF from DDH). Let G be a cyclic group of prime order q generated by g 2 G. Let H : M ! G be a hash function, which we shall model as a random oracle (see Section 8.10.2). Let F be the PRF defined over (Zq , M, G) as follows: F (k, m) := H(m)k for k 2 Zq , m 2 M. 438
Show that F is a secure PRF in the random oracle model for H under the DDH assumption for G. Hint: Use the results of Exercises 10.6 and 10.7. ˆ : G ⇥ G ! Y be a hash 11.2 (Simple PRF from CDH). Continuing with Exercise 11.1, let H ˆ function, which we again model as a random oracle. Let F be the PRF defined over (Zq , M, Y) as follows: ⌘ ⇣ ˆ H(m), H(m)k for k 2 Zq , m 2 M. Fˆ (k, m) := H ˆ under the CDH assumption Show that Fˆ is a secure PRF in the random oracle model for H and H for G. Hint: Use the result of Exercise 10.4. 11.3 (Oblivious PRF from DDH). Your proof that the PRF F F presented in Exercise 11.1 should still go through even if the value g k is publicly known. Using this fact, we can design a protocol that allows F to be evaluated obliviously. This means that if Bob has a key k and Alice has an input m, there is a simple protocol that lets Alice compute F (k, m) in such a way that Bob does not learn anything about m and Alice learns nothing about k besides F (k, m) and g k . Hint: Alice starts by sending Bob H(m) · g ⇢ for random ⇢ 2 Zq — see also Exercise 10.4. 11.4 (Broken variant of RSA). Consider the following broken version of the RSA publickey encryption scheme: key generation is as in ERSA , but to encrypt a message m 2 Zn with public key pk = (n, e) do E(pk , m) := me . Decryption is done using the RSA trapdoor. Clearly this scheme is not semantically secure. Even worse, suppose one encrypts a random message m 2 {0, 1, . . . , 264 } to obtain c := me mod n. Show that for 35% of plaintexts in [0, 264 ], an adversary can recover the complete plaintext m from c using only 235 eth powers in Zn . Hint: Use the fact that about 35% of the integers m in [0, 264 ] can be written as m = m1 · m2 where m1 , m2 2 [0, 234 ]. 11.5 (Multiplicative ElGamal). Let G be a cyclic group of prime order q generated by g 2 G. Consider a simple variant of the ElGamal encryption system EMEG = (G, E, D) that is defined over (G, G2 ). The key generation algorithm G is the same as in EEG , but encryption and decryption work as follows: • for a given public key pk = u 2 G and message m 2 G: E(pk , m) :=
R
Zq , v
g , c
m · u , output (v, c)
• for a given secret key sk = ↵ 2 Zq and a ciphertext (v, c) 2 G2 : D(sk , (v, c) ) :=
output c/v ↵
(a) Show that EMEG is semantically secure assuming the DDH assumption holds in G. In particular, you should show that the advantage of any adversary A in breaking the semantic security of EMEG is bounded by 2✏, where ✏ is the advantage of an adversary B (which is an elementary wrapper around A) in the DDH attack game. (b) Show that EMEG is not semantically secure if the DDH assumption does not hold in G. 439
(c) Show that EMEG has the following property: given a public key pk , and two ciphertexts c1 R E(pk , m1 ) and c2 R E(pk , m2 ), it is possible to create a new ciphertext c which is an encryption of m1 · m2 . This property is called a multiplicative homomorphism. 11.6 (An attack on multiplicative ElGamal). Let p and q be large primes such that q divides p 1. Let G be the order q subgroup of Z⇤p generated by g 2 G and assume that the DDH assumption holds in G. Suppose we instantiate the ElGamal system from Exercise 11.5 with the group G. However, plaintext messages are chosen from the entire group Z⇤p so that the system is defined over (Z⇤p , G ⇥ Z⇤p ). Show that the resulting system is not semantically secure. 11.7 (Extending the message space). Suppose that we have a publickey encryption scheme E = (G, E, D) with message space M. From this, we would like to build an encryption scheme with message space M2 . To this end, consider the following encryption scheme E 2 = (G2 , E 2 , D2 ), where G2 () E 2 pk , (m0 , m1 ) 2
D sk , (c0 , c1 )
:=
(pk 0 , sk 0 )
R
G(),
(pk 1 , sk 1 )
R
G(), output pk := (pk 0 , pk 1 ) and sk := (sk 0 , sk 1 )
:=
E(pk 0 , m0 ), E(pk 1 , m1 )
:=
D(sk 0 , c0 ), D(sk 1 , c1 )
Show that E 2 is semantically secure, assuming E itself is semantically secure. 11.8 (Modular hybrid construction). Both of the encryption schemes presented in this chapter, ETDF in Section 11.4 and EEG in Section 11.5, as well as many other schemes used in practice, have a “hybrid” structure that combines an asymmetric component and a symmetric component in a fairly natural and modular way. The symmetric part is, of course, the symmetric cipher Es = (Es , Ds ), defined over (K, M, C). The asymmetric part can be understood in abstract terms as what is called a key encapsulation mechanism, or KEM. A KEM Ekem consists of a tuple of algorithms (G, Ekem , Dkem ). Algorithm G is invoked as (pk , sk ) R G(). Algorithm Ekem is invoked as (k, ckem ) R Ekem (pk ), where k 2 K and ckem 2 Ckem . Algorithm Dkem is invoked as k Dkem (sk , ckem ), where k 2 K [ {reject} and ckem 2 Ckem . We say that Ekem is defined over (K, Ckem ). We require that Ekem satisfies the following correctness requirement: for all possible outputs (pk , sk ) of G(), and all possible outputs (k, ckem ) of Ekem (pk ), we have Dkem (sk , ckem ) = k. We can define a notion of semantic security in terms of an attack game between a challenger and an adversary A, as follows. In Experiment b, for b = 0, 1, the challenger computes (pk , sk )
R
G(), (k0 , ckem )
R
Ekem (pk ), k1
R
K,
and sends (kb , ckem ) to A. Finally, A outputs ˆb 2 {0, 1}. As usual, if Wb is the event that A outputs 1 in Experiment b, we define A’s advantage with respect to Ekem as SSadv[A, Ekem ] := Pr[W0 ] Pr[W1 ], and if this advantage is negligible for all efficient adversaries, we say that Ekem is semantically secure. Now consider the hybrid publickey encryption scheme E = (G, E, D), constructed out of Ekem and Es , and defined over (M, Ckem ⇥ C). The key generation algorithm for E is the same as that of Ekem . The encryption algorithm E works as follows: E(pk , m) :=
(k, ckem )
R
Ekem (pk ), c 440
R
Es (k, m), output (ckem , c)
.
The decryption algorithm D works as follows: D(sk , (ckem , c)) :=
m
reject, k
output m
.
Dkem (sk , ckem ), if k 6= reject then m
Ds (k, c),
(a) Prove that E satisfies the correctness requirement for a public key encryption scheme, assuming Ekem and Es satisfy their corresponding correctness requirements. (b) Prove that E is semantically secure, assuming that Ekem and Es are semantically secure. You should prove a concrete security bound that says that for every adversary A attacking E, there are adversaries Bkem and Bs (which are elementary wrappers around A) such that SSadv[A, E] 2 · SSadv[Bkem , Ekem ] + SSadv[Bs , Es ]. (c) Describe the KEM corresponding to ETDF and prove that it is semantically secure (in the random oracle model, assuming T is one way). (d) Describe the KEM corresponding to EEG and prove that it is semantically secure (in the random oracle model, under the CDH assumption for G). (e) Let Ea = (G, Ea , Da ) be a publickey encryption scheme defined over (K, Ca ). Define the KEM Ekem = (G, Ekem , Da ), where Ekem (pk ) :=
k
R
K, ckem
R
Ea (pk , k), output (k, ckem )
.
Show that Ekem is semantically secure, assuming that Ea is semantically secure. Discussion: Part (e) shows that one can always build a KEM from a publickey encryption scheme by just using the encryption scheme to encrypt a symmetric key; however, parts (c) and (d) show that there are more direct and efficient ways to do this. 11.9 (Multikey CPA security). Generalize the definition of CPA security for a publickey encryption scheme to the multikey setting. In this attack game, the adversary gets to obtain encryptions of many messages under many public keys. Show that semantic security implies multikey CPA security. You should show that security degrades linearly in Qk Qe , where Qk is a bound on the number of keys, and Qe is a bound on the number of encryption queries per key. That is, the advantage of any adversary A in breaking the multikey CPA security of a scheme is at most Qk Qe · ✏, where ✏ is the advantage of an adversary B (which is an elementary wrapper around A) that breaks the scheme’s semantic security. 11.10 (A tight reduction for multiplicative ElGamal). We proved in Exercise 11.9 that semantic security for a publickey encryption scheme implies multikey CPA security; however, the security degrades significantly as the number of keys and encryptions increases. Consider the multiplicative ElGamal encryption scheme EMEG from Exercise 11.5. You are to show show a tight reduction from multikey CPA security for EMEG to the DDH assumption, which does not degrade at all as the number of keys and encryptions increases. In particular, you should show that the advantage of any adversary A in breaking the multikey CPA security of EMEG is bounded by 2(✏ + 1/q), where ✏ is the advantage of an adversary B (which is an elementary wrapper around A) in the DDH attack game. 441
Note: You should assume that in the multikey CPA game, the same group G and generator g 2 G is used throughout. Hint: Use the results of Exercises 10.6, 10.7, and 10.8. 11.11 (An easy discretelog group). Let n be a large integer and consider the following subset of Z⇤n2 : Gn := [an + 1]n2 2 Z⇤n2 : a 2 {0, . . . , n 1} (a) Show that Gn is a multiplicative subgroup of Z⇤n2 of order n.
(b) Which elements of Gn are generators? (c) Choose an arbitrary generator g 2 Gn and show that the discretelog problem in Gn is easy. 11.12 (Pallier encryption). Let us construct another publickey encryption scheme (G, E, D) that makes use of RSA composites: • The key generation algorithm is parameterized by a fixed value ` and runs as follows: G(`) :=
generate two distinct random `bit primes p and q, n pq, d (p 1)(q 1)/2 pk n, sk d output (pk , sk )
• for a given public key pk = n and message m 2 {0, . . . , n encryption algorithm runs as follows: E(pk , m) :=
h
R
Z⇤n2 ,
c
R
g m hn 2 Z⇤n2 ,
1}, set g := [n + 1]n2 2 Z⇤n2 . The
output c.
(a) Explain how the decryption algorithm D(sk , c) works. Hint: Using the notation of Exercise 11.11, observe that cd falls in the subgroup Gn which has an easy discretelog. (b) Show that this publickey encryption scheme is semantically secure under the following assumption: let n be a product of two random `bit primes, let u be uniform in Z⇤n2 , let v be uniform in the subgroup (Zn2 )n := {hn : h 2 Z⇤n2 }, then the distribution (n, u) is computationally indistinguishable from the distribution (n, v). Discussion: This encryption system, called Pallier encryption, has a useful property called an additive homomorphism: for ciphertexts c0 R E(pk , m0 ) and c1 R E(pk , m1 ), the product c c0 · c1 is an encryption of m0 + m1 mod n. 11.13 (Hash DiffieHellman). Let G be a cyclic group of prime order q generated by g 2 G. Let H : G ! K be a hash function. We say that the Hash DiffieHellman (HDH) assumption holds for (G, H) if the distribution g ↵ , g , H(g ↵ ) is computationally indistinguishable from the distribution (g ↵ , g , k) where ↵, R Zq and k R K. 442
(a) Show that if H is modeled as a random oracle and the CDH assumption holds for G, then the HDH assumption holds for (G, H). (b) Show that if H is a secure KDF and the DDH assumption holds for G, then the HDH assumption holds for (G, H). (c) Prove that the ElGamal publickey encryption scheme EEG is semantically secure if the HDH assumption holds for (G, H). 11.14 (Anonymous publickey encryption). Suppose t people publish their publickeys pk 1 , . . . , pk t . Alice sends an encrypted message to one of them, say pk 5 , but she wants to ensure that no one (other than user 5) can tell which of the t users is the intended recipient. You may assume that every user, other than user 5, who tries to decrypt Alice’s message with their secret key, obtains fail. (a) Define a security model that captures this requirements. The adversary should be given t public keys pk 1 , . . . , pk t and it then selects the message m that Alice sends. Upon receiving a challenge ciphertext, the adversary should learn nothing about which of the t public keys is the intended recipient. A system that has this property is said to be an anonymous publickey encryption scheme. (b) Show that the ElGamal publickey encryption system EEG is anonymous. (c) Show that the RSA publickey encryption system ERSA is not anonymous. Assume that all t public keys are generated using the same RSA parameters ` and e. 11.15 (Access structures). Generalize the ElGamal threshold decryption scheme of Section 11.6.2 to the following settings: The s key servers are split into two disjoint groups S1 and S2 , and decryption should be possible only if the combiner receives at least t1 responses from the set S1 , and at least t2 responses from the set S2 , where t1 S1  and t2 S2 . Adapt the security definition to these settings, and prove that your scheme is secure. Discussion: An access structure is the set of subsets of {0, . . . , s 1} that should be able to decrypt. In Section 11.6.2 we looked at a threshold access structure, and this exercise looks at a slightly more general threshold access structure. Other access structures can be achieved using more general secret sharing schemes, as long as the secret is reconstructed using a linear function of the given shares. Such schemes, called linear secret sharing schemes (LSSS), are surveyed in [5]. 11.16 (RSA threshold decryption). Let us show how to enable simple threshold decryption for the RSA public key encryption scheme of Section 11.4.1. (a) Recall that the key generation algorithm generates numbers n, e, d, where n is the RSA modulus, e is the encryption exponent, and d is the decryption exponent. We extend the key generation algorithm with two more steps: choose a random integer d1 in [1, n2 ] and set d2 = d1 d 2 Z. Then output the two key shares sk 1 := (n, d1 ) and sk 2 := (n, d2 ), and the public key pk := (n, e). Explain how to use this setup for 2outof2 threshold decryption, to match the framework of Definition 11.6. Hint: Show that the distribution of the key share d2 is statistically close to the uniform distribution on {1, . . . , n2 }. 443
(b) Prove that your scheme from part (a) satisfies the security definition for 2outof2 threshold decryption (Definition 11.9). (c) Generalize the scheme to provide 2outof3 threshold decryption, using the mechanism of Exercise 2.20. Prove that the scheme is secure. 11.17 (Proxy reencryption). Bob works for the Acme corporation and publishes a publickey pk bob so that all incoming emails to Bob are encrypted under pk bob . When Bob goes on vacation he instructs the company’s mail server to forward all his incoming encrypted email to Alice. Alice’s public key is pk alice . The mail server needs a way to translate an email encrypted under publickey pk bob into an email encrypted under publickey pk alice . This would be easy if the mail server had sk bob , but then the mail server can read all of Bob’s incoming email. Suppose that pk bob and pk alice are public keys for the ElGamal encryption scheme EEG discussed in Section 11.5, both based on the same group G with generator g 2 G. Then the mail server can do the translation from pk bob to pk alice while learning nothing about the email contents. 0
(a) Suppose pk alice = g ↵ and pk bob = g ↵ . Show that giving ⌧ := ↵/↵0 to the mail server lets it translate an email encrypted under pk bob into an email encrypted under pk alice , and viceversa. (b) Assume that EEG is semantically secure. Show that the adversary cannot break semantic 0 security for Alice, even if it is given Bob’s public key g ↵ along with the translation key ⌧ . 11.18 (A voting system). Consider an election system where voters vote for one of two parties and their vote is either 0 and 1. The election service publishes an ElGamal publickey pk and every voter sends to the election service its vote bi 2 {0, 1}, encoded as the group element g bi , encrypted under pk using the multiplicative ElGamal system from Exercise 11.5. The election service needs to determine how many people voted 0 and how many voted 1. This is equivalent to computing Pn S := i=1 bi where n is the total number of voters who sent in their encrypted votes. You may assume that n is at most 109 . (a) Suppose the election service is partitioned into two components, a tabulation service and a decryption authority. Incoming votes are received by the tabulation service and the decryption authority is an o✏ine box that holds sk and only communicates with the tabulation service. Show that the tabulation service can send a single ElGamal ciphertext c⇤ to the decryption authority who then decrypts c⇤ and outputs S in the clear. If both parties are honestly following your protocol then neither one learns anything other than S about the individual votes. Explain how the tabulation service constructs c⇤ . Hint: Use Exercise 11.5 part (c). (b) Show that a single malicious voter can make S come out to be whatever value that voter wants. Discussion: While part (b) shows that this voting system is insecure as is, this idea can form the basis of a secure election system. See [28] for details.
444
Chapter 12
Chosen ciphertext secure public key encryption In Chapter 11, we introduced the notion of publickey encryption. We also defined a basic form of security called semantic security, which is completely analogous to the corresponding notion of semantic security in the symmetrickey setting. We observed that in the publickey setting, semantic security implies security against a chosen plaintext attack, i.e., CPA security. In this chapter, we study the stronger notion of security against chosen ciphertext attack, or CCA security. In the CPA attack game, the decryption key is never used, and so CPA security provides no guarantees in any realworld setting in which the decryption key is actually used to decrypt messages. The notion of CCA security is designed to model a wide spectrum of realworld attacks, and it is considered the “gold standard” for security in the publickey setting. We briefly introduced the notion of CCA security in the symmetrickey setting in Section 9.2, and the definition in the publickey setting is a straightforward translation of the definition in the symmetrickey setting. However, it turns out CCA security plays a more fundamental role in the publickey setting than in the symmetrickey setting.
12.1
Basic definitions
As usual, we formulate this notion of security using an attack game, which is a straightforward adaptation of the CCA attack game in the symmetric settings (Attack Game 9.2) to the publickey setting. Attack Game 12.1 (CCA security). For a given publickey encryption scheme E = (G, E, D), defined over (M, C), and for a given adversary A, we define two experiments. Experiment b
(b = 0, 1):
• The challenger computes (pk , sk )
R
G() and sends pk to the adversary.
• A then makes a series of queries to the challenger. Each query can be one of two types: – Encryption query: for i = 1, 2, . . . , the ith encryption query consists of a pair of messages (mi0 , mi1 ) 2 M2 , of the same length. The challenger computes ci R E(pk , mib ) and sends ci to A. 445
– Decryption query: for j = 1, 2, . . . , the jth decryption query consists of a ciphertext cˆj 2 C that is not among the responses to the previous encryption queries, i.e., / {c1 , c2 , . . .}. cˆj 2 The challenger computes m ˆj
D(sk , cˆj ), and sends m ˆ j to A.
• At the end of the game, the adversary outputs a bit ˆb 2 {0, 1}. Let Wb is the event that A outputs 1 in Experiment b and define A’s advantage with respect to E as CCAadv[A, E] := Pr[W0 ] Pr[W1 ] . 2 Definition 12.1 (CCA Security). A publickey encryption scheme E is called semantically secure against a chosen ciphertext attack, or simply CCA secure, if for all efficient adversaries A, the value CCAadv[A, E] is negligible. Just as we did in the symmetrickey setting, we can consider a restricted attack game in which the adversary makes only a single encryption query: Definition 12.2 (1CCA security). In Attack Game 12.1, if the adversary A is restricted to making a single encryption query, we denote its advantage by 1CCAadv[A, E]. A publickey encryption scheme E is onetime semantically secure against chosen ciphertext attack, or simply, 1CCA secure, if for all efficient adversaries A, the value 1CCAadv[A, E] is negligible. Notice that if we strip away the decryption queries, 1CCA security corresponds to semantic security, and CCA security corresponds to CPA security. We showed in Theorem 11.1 that semantic security for a publickey encryption scheme implies CPA security. A similar result holds with respect to chosen ciphertext security, namely, that 1CCA security implies CCA security. Theorem 12.1. If a publickey encryption scheme E is 1CCA secure, then it is also CCA secure. In particular, for every CCA adversary A that plays Attack Game 12.1 with respect to E, and which makes at most Qe encryption queries to its challenger, there exists a 1CCA adversary B as in Definition 12.2, where B is an elementary wrapper around A, such that CCAadv[A, E] = Qe · 1CCAadv[B, E].
The proof is a simple hybrid argument that is almost identical to that of Theorem 11.1, and we leave the details as an easy exercise to the reader. Using another level of hybrid argument, one can also extend this to the multikey setting as well — see Exercise 12.5. Since 1CCA security implies CCA security, if we want to prove that a particular publickey encryption scheme is CCA secure, we will typically simply prove 1CCA security. So it will be helpful to study the 1CCA attack game in a bit more detail. We can view the 1CCA attack game as proceeding in a series of phases: Initialization phase: the challenger generates (pk , sk )
R
G() and sends pk to the adversary.
Phase 1: the adversary submits a series of decryption queries to the challenger; each such query is a ciphertext cˆ 2 C, to which the challenger responds with m ˆ D(sk , cˆ). 446
Encryption query: the adversary submits a single encryption query (m0 , m1 ) to the challenger; in Experiment b (where b = 0, 1), the challenger responds with c R E(pk , mb ). Phase 2: the adversary again submits a series of decryption queries to the challenger; each such query is a ciphertext cˆ 2 C, subject to the restriction that cˆ 6= c, to which the challenger responds with m ˆ D(sk , cˆ). Finish: at the end of the game, the adversary outputs a bit ˆb 2 {0, 1}. As usual, as discussed in Section 2.3.5, Attack Game 12.1 can be recast as a “bit guessing” game, where instead of having two separate experiments, the challenger chooses b 2 {0, 1} at random, and then runs Experiment b against the adversary A. In this game, we measure A’s bitguessing advantage CCAadv⇤ [A, E] (and 1CCAadv⇤ [A, E]) as Pr[ˆb = b] 1/2. The general result of Section 2.3.5 applies here as well: CCAadv[A, E] = 2 · CCAadv⇤ [A, E].
(12.1)
And similarly, for adversaries restricted to a single encryption query, we have: 1CCAadv[A, E] = 2 · 1CCAadv⇤ [A, E].
12.2
(12.2)
Understanding CCA security
The definition of CCA security may seem rather unintuitive at first. Indeed, one might ask: in the attack game, why can the adversary get any message decrypted except the ones he really wants to decrypt? One answer is that without this restriction, it would be impossible to satisfy the definition. However, this is not a very satisfying answer, and it begs the question as to whether the entire definitional framework makes sense. In this section, we explore the definition of CCA security from several angles. Hopefully, by the end, the reader will understand why this definition makes sense, and what it is good for.
12.2.1
CCA security and ciphertext malleability
Our first example illustrates an important property of CCA secure systems: they are nonmalleable. That is, given an encryption c of some message m, the attacker cannot create a di↵erent ciphertext c0 that decrypts to a message m0 that is somehow related to m. The importance of this will become clear in the example below. Consider a professor, Bob, who collects homework by email. Moreover, assume that Bob generates a public key/secret key pair (pk , sk ) for a publickey encryption scheme, and gives pk to all of his students. When a student Alice submits an email, she encrypts it under pk . To make things concrete, suppose that the publickey encryption scheme is the semantically secure scheme ETDF presented in Section 11.4, which is based on a trapdoor function along with some symmetric cipher Es . The only requirement on Es is that it is semantically secure, so let us assume that Es is a stream cipher (such as AES in counter mode). When Alice encrypts the email message m containing her homework using ETDF and pk , the resulting ciphertext is of the form (y, c), where y = F (pk , x) and c = G(H(x)) m. Here, H is a hash function and G is a PRG. 447
As we saw in Section 3.3.2, any stream cipher is extremely malleable, and the publickey scheme ETDF inherits this weakness. In particular, an attacker Molly can do essentially the same thing here as she did in Section 3.3.2. Namely, assuming that Alice’s email message m starts with the header From:Alice, by flipping a few bits of the symmetrickey ciphertext c, Molly obtains another ciphertext c0 that decrypts (under the same symmetric key) to a message m0 that is identical to m, except that the header now reads From:Molly. Using the above technique, Molly can “steal” Alice’s homework as follows. She intercepts Alice’s ciphertext (y, c). She then modifies the symmetrickey ciphertext c to obtain c0 as above, and sends the publickey ciphertext (y, c0 ) to Bob. Now, when Professor Bob decrypts (y, c0 ), he will essentially see Alice’s homework, but Bob will mistakenly think that the homework was submitted by Molly, and give Molly credit for it. The attack described so far is a good example of a chosen ciphertext attack, which could not succeed if the publickey encryption scheme were actually CCA secure. Indeed, if given (y, c) it is possible for Molly to create a new ciphertext (y, c0 ) where the header From:Alice is changed to From:Molly, then the system cannot be CCA secure. For such a system, we can design a simple CCA adversary A that has advantage 1 in the CCA security game. Here is how. • Create a pair of messages, each with the same header, but di↵erent bodies. Our adversary A submits this pair as an encryption query, obtaining (y, c). • A then uses Molly’s algorithm to create a ciphertext (y, c0 ), which should encrypt a message with a di↵erent header but the same body. • A then submits (y, c0 ) as a decryption query, and outputs 0 or 1, depending on which body it sees. As we have shown, if Alice encrypts her homework using a CCAsecure system, she is assured that no one can steal her homework by modifying the ciphertext she submitted. CCA security, however, does not prevent all attacks on this homework submission system. An attacker can maliciously submit a homework on behalf of Alice, and possibly hurt her grade in the class. Indeed, anyone can send an encrypted homework to the professor, and in particular, a homework that begins with From:Alice. Preventing this type of attack requires tools that we will develop later. In Section 13.7, where we develop the notion of signcryption, which is one way to prevent this attack.
12.2.2
CCA security vs authentication
When we first encountered the notion of CCA security in the symmetrickey setting, back in Section 9.2, we saw that CCA security was implied by AE security, i.e., ciphertext integrity plus CPA security. Moreover, we saw that ciphertext integrity could be easily added to any CPAsecure encryption scheme using the encryptthenMAC method. We show here that this does not work in the publickey setting: simply adding an authentication wrapper does not make the system CCA secure. Consider again the homework submission system example in the previous section. If we start with a scheme, like ETDF , which is not itself CCA secure, we might hope to make it CCA secure using encryptthenMAC: Alice wraps the ciphertext (y, c) with some authentication data computed from (y, c). Say, Alice computes a MAC tag t over (y, c) using a secret key that she shares with Bob and sends (y, c, t) to Bob (or, instead of a MAC, she computes a digital signature on (y, c), a concept 448
discussed in Chapter 13). Bob can check the authentication data to make sure the ciphertext was generated by Alice. However, regardless of the authentication wrapper used, Molly can still carry out the attack described in the previous section. Here is how. Molly intercepts Alice’s ciphertext (y, c, t), and computes (y, c0 ) exactly as before. Now, since Molly is a registered student in Bob’s course, she presumably is using the same authentication mechanism as all other students, so she simply computes her own authentication tag t0 on ciphertext (y, c0 ) and sends (y, c0 , t0 ) to Bob. Bob receives (y, c0 , t0 ), and believes the authenticity of the ciphertext. When Bob decrypts (y, c0 ), the header From:Molly will look perfectly consistent with the authentication results. What went wrong? Why did the strategy of authenticating ciphertexts provide us with CCA security in the symmetrickey setting, but not in the publickey setting? The reason is simply that in the publickey setting, anyone is allowed to send an encrypted message to Bob using Bob’s public key. The added flexibility that publickey encryption provides makes it more challenging to achieve CCA security, yet CCA security is vital for security in realworld systems. (We will discuss in detail how to securely combine CCAsecure publickey encryption and digital signatures when we discuss signcryption in Section 13.7.)
12.2.3
CCA security and key escrow
Consider again the key escrow example discussed in Section 11.1.2. Recall that in that example, Alice encrypts a file f using a symmetric key k. Among other things, Alice stores along with the encrypted file an escrow of the file’s encryption key. Here, the escrow is an encryption cES of k under the public key of some escrow service. If Alice works for some company, then if need be, Alice’s manager or other authorized entity can retrieve the file’s encryption key by presenting cES to the escrow service for decryption. If the escrow service uses a CCAsecure encryption scheme, then it is possible to implement an access control policy which can mitigate against potential abuse. This can be done as follows. Suppose that in forming the escrowciphertext cES , Alice encrypts the pair (k, h) under the escrow service’s public key, where h is a collisionresistant hash of the metadata md associated with the file f : this might include the name of the file, the time that it was created and/or modified, and perhaps the identity of the owner of the file (Alice, in this case). Let us also assume that all of this metadata md is stored on the file system in the clear along with the encrypted file. Now suppose a requesting entity presents the escrowciphertext cES to the escrow service, along with the corresponding metadata md . The escrow service may impose some type of access control policy, based on the given metadata, along with the identity or credentials of the requesting entity. Such a policy could be very specific to a particular company or organization. For example, the requesting entity may be Alice’s manager, and it is company policy that Alice’s manager should have access to all files owned by Alice. Or the requesting entity may be an external auditor that is to have access to all files created by certain employees on a certain date. To actually enforce this access control policy, not only must the escrow service verify that the requesting identity’s credentials and the supplied metadata conform to the access control policy, the escrow service must also perform the following check: after decrypting the escrowciphertext cES to obtain the pair (k, h), it must check that h matches the hash of the metadata supplied by the requesting entity. Only if these match does the escrow service release the key k to the requesting entity. This type of access control can prevent certain abuses. For example, consider the external auditor who has the right to access all files created by certain employees on a certain date. Suppose 449
the auditor himself is a bit too nosy, and during the audit, wants to find out some information in a personal file of Alice that is not one of the files targeted by the audit. The above implementation of the escrow service, along with CCA security, ensures that the nosy auditor cannot obtain this unauthorized information. Indeed, suppose cES is the escrowciphertext associated with Alice’s personal file, which is not subject to the audit, and that this file has metadata md . Suppose the auditor submits a pair (c0ES , md 0 ) to the escrow service. There are several cases to consider: • if md 0 = md , then the escrow service will reject the request, as the metadata md of Alice’s personal file does not fit the profile of the audit; • if md 0 6= md and c0ES = cES , then the collision resistance of the hash ensures that the escrow service will reject the request, as the hash embedded in the decryption of c0ES will not not match the hash of the supplied metadata md 0 ; • if md 0 6= md and c0ES 6= cES , then the escrow service may or may not accept the request, but even if it does, CCA security and the fact that c0ES 6= cES ensures that no information about the encryption key for Alice’s personal file is revealed. This implementation of an escrow service is pretty good, but it is far from perfect: • It assumes that Alice follows the protocol of actually encrypting the file encryption key along with the correct metadata. Actually, this may not be such an unreasonable assumption, as these tasks will be performed automatically by the file system on Alice’s behalf, and so it may not be so easy for a misbehaving Alice to circumvent this protocol. • It assumes that the requesting entity and the escrow service do not collude. Treating the metadata as associated data. In Section 12.7 we define publickey encryption with associated data, which is the publickey analogue of symmetric encryption with associated data from Section 9.5. Here the publickey encryption and decryption algorithms take a third input called associated data. The point is that decryption reveals no useful information if the given associated data used in decryption is di↵erent from the one used in encryption. The metadata information md in the escrow system above can be treated as associated data, instead of appending it to the plaintext. This will result in a smaller ciphertext while achieving the same security goals. In fact, associating metadata to a ciphertext for the purpose described above is a very typical application of associated data in a publickey encryption scheme.
12.2.4
Encryption as an abstract interface
To conclude our motivational discussion of CCA security we show that it abstractly captures a “correct” and very natural notion of security. We do this by describing encryption as an abstract interface, as discussed in Section 9.3 in the symmetric case. The setting is as follows. We have a sender S and receiver R, who are participating in some protocol, during which S drops messages m1 , m2 , . . . into his outbox, and R retrieves messages from his inbox. While S and R do not share a secret key, we assume that R has generated public key/secret key pair (pk , sk ), and that S knows R’s public key pk . That is the abstract interface. In a real implementation, when mi is placed in S’s outbox, it is encrypted under pk , yielding a corresponding ciphertext ci , which is sent over the wire to R. On 450
the receiving end, when a ciphertext cˆ is received at R’s end of the wire, it is decrypted using sk , and if the decryption is a message m ˆ 6= reject, the message m ˆ is placed in R’s inbox. Note that while we are syntactically restricting ourselves to a single sender S, this restriction is superficial: in system with many users, all of them have access to R’s public key, and so we can model such a system by allowing all users to place messages in S’s outbox. Just as in Section 9.3, an attacker may attempt to subvert communication in several ways: • The attacker may drop, reorder, or duplicate the ciphertexts sent by S. • The attacker may modify ciphertexts sent by S, or inject ciphertexts computed in some arbitrary fashion. • The attacker may have partial knowledge — or even influence the choice — of the messages sent by S. • The attacker can obtain partial knowledge of some of the messages retrieved by R, and determine if a given ciphertext delivered to R was rejected. We now describe an ideal implementation of this interface. It is slightly di↵erent from the ideal implementation in Section 9.3 — in that section, we were working with the notion of AE security, while here we are working with the notion of CCA security. When S drops mi in its outbox, instead of encrypting mi , the ideal implementation creates a ciphertext ci by encrypting a dummy message dummy i , that has nothing to do with mi (except that it should be of the same length). Thus, ci serves as a “handle” for mi , but does not contain any information about mi (other than its length). When ci arrives at R, the corresponding message mi is magically copied from S’s outbox to R’s inbox. If a ciphertext cˆ arrives at R that is not among the previously generated ci ’s, the ideal implementation decrypts cˆ using sk as usual. CCA security implies that this ideal implementation of the service is for all practical purposes equivalent to the real implementation. In the ideal implementation, we see that messages magically jump from S to R, in spite of any information the adversary may glean by getting R to decrypt other ciphertexts — the ciphertexts generated by S in the ideal implementation serve simply as handles for the corresponding messages, but do not carry any other useful information. Hopefully, analyzing the security properties of a higherlevel protocol will be much easier using this ideal implementation. Note that even in the ideal implementation, the attacker may still drop, reorder, or duplicate ciphertexts, and these will cause the corresponding messages to be dropped, reordered, or duplicated. A higherlevel protocol can easily take measures to deal with these issues. We now argue informally that when E is CCA secure, the real world implementation is indistinguishable from the ideal implementation. The argument is similar to that in Section 9.3. It proceeds in two steps, starting with the real implementation, and in each step, we make a slight modification. • First, we modify the real implementation of R’s inbox, as follows. When a ciphertext cˆ arrives on R’s end, the list of ciphertexts c1 , c2 , . . . previously generated by S is scanned, and if cˆ = ci , then the corresponding message mi is magically copied from S’s outbox into R’s inbox, without actually running the decryption algorithm. The correctness property of E ensures that this modification behaves exactly the same as the real implementation. Note that in this modification, any ciphertext that arrives at R’s end 451
that is not among the ciphertexts previously generated by S will be decrypted as usual using sk . • Second, we modify the implementation of S’s outbox, replacing the encryption of mi with the encryption of dummy i . The implementation of R’s inbox remains as in the first modification. Here is where we use the CCA security property: if the attacker could distinguish the second modification from the first, we could use the attacker to break the CCA security of E. Since the second modification is identical to the ideal implementation, we see that the real and ideal implementations are indistinguishable from the adversary’s point of view. Just as in Section 9.3, we have ignored the possibility that the ci ’s generated by S are not unique. Certainly, if we are going to view the ci ’s as handles in the ideal implementation, uniqueness would seem to be an essential property. Just as in the symmetric case, CPA security (which is implied by CCA security) guarantees that the ci ’s are unique with overwhelming probability (the reader can verify that the result of Exercise 5.11 holds in the publickey setting as well).
12.3
CCAsecure encryption from trapdoor function schemes
We now turn to constructing CCAsecure publickey encryption schemes. We begin with a construction from a general trapdoor function scheme satisfying certain properties. We use this to obtain a CCAsecure system from RSA. Later, in Section 12.6, we will show how to construct suitable trapdoor functions (in the random oracle model) from arbitrary, CPAsecure publickey encryption schemes. Using the result in this section, all these trapdoor functions give us CCAsecure encryption schemes. Consider again the publickey encryption scheme ETDF = (G, E, D) discussed in Section 11.4, which is based on an arbitrary trapdoor function scheme T = (G, F, I), defined over (X , Y). Let us briefly recall this scheme: it makes use of a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), and a hash function H : X ! K, which we model as a random oracle. The message space for ETDF is M and the ciphertext space is Y ⇥ C. The key generation algorithm for ETDF is the same as the key generation algorithm for T , and encryption and decryption work as follows: E(pk , m)
:=
x R X, y F (pk , x), k output (y, c);
D(sk , (y, c) )
:=
x I(sk , y), k output m.
H(x), c
H(x), m
R
Es (k, m)
Ds (k, c)
If X = 6 Y, that is, if T is not a trapdoor permutation scheme, we have to modify the scheme slightly to get a scheme that is CCA secure. Basically, we modify the decryption algorithm to explicitly check that the given value y 2 Y is actually in the image of F (pk , ·). So the scheme we 0 will analyze is ETDF = (G, E, D0 ), where D0 (sk , (y, c) )
:=
x I(sk , y) if F (pk , x) = y then k H(x), m else m reject output m. 452
Ds (k, c)
0 We will prove that ETDF is CCA secure if we model H as a random oracle, under appropriate assumptions. The first assumption we will make is that Es is 1CCA secure (see Section 9.6). We also have to assume that T is oneway. However, when X 6= Y, we need a somewhat stronger assumption: that T is oneway even given access to an “image oracle”. Essentially, this means that given pk and y = F (pk , x) for randomly chosen x 2 X , it is hard to compute x, even given access to an oracle that will answer arbitrary questions of the form “does a given yˆ 2 Y lie in the image of F (pk , ·)?”. We formalize this notion by giving an attack game that is similar to Attack Game 10.2, but where the adversary has access to an image oracle.
Attack Game 12.2 (Oneway trapdoor function scheme even with image oracle). For a given trapdoor function scheme T = (G, F, I), defined over (X , Y), and a given adversary A, the attack game runs as follows: • The challenger computes (pk , sk )
R
G(),
x
R
X,
y
F (pk , x)
and sends (pk , y) to the adversary. • The adversary makes a series of image oracle queries to the challenger. Each such query is of the form yˆ 2 Y, to which the challenger replies “yes” if F (pk , I(sk , yˆ)) = yˆ, and “no” otherwise. • The adversary outputs x ˆ 2 X. We define the adversary’s advantage in inverting T given access to an image oracle, denoted IOWadv[A, T ], to be the probability that x ˆ = x. 2 Definition 12.3. We say that a trapdoor function scheme T is one way given an image oracle if for all efficient adversaries A, the quantity IOWadv[A, T ] is negligible. In Exercise 12.13 we show that (in the random oracle model) every one way trapdoor function scheme can be easily converted into one that is one way given an image oracle. 0 , assuming T is oneway given an image oraThe next theorem proves the CCA security of ETDF cle, Es is 1CCA secure (see Definition 9.6), and H is modeled as a random oracle. In Exercise 12.12 we explore an alternative analysis of this scheme under di↵erent assumptions. 0 In proving this theorem, we just prove that ETDF is 1CCA secure (see Definition 12.2). By virtue of Theorem 12.1, this is sufficient. Recall that in the random oracle model (see Section 8.10), the function H is modeled as a random function O chosen at random from the set of all functions Funs[X , K]. This means that in the random oracle version of the 1CCA attack game, the challenger chooses O at random. In any computation where the challenger would normally evaluate H, it evaluates O instead. In addition, the adversary is allowed to ask the challenger for the value of the function O at any point of its choosing. The adversary may make any number of such “random oracle queries” at any time of its choosing, arbitrarily interleaved with its usual encryption and 0 0 decryption queries. We use 1CCAro adv[A, ETDF ] to denote A’s advantage against ETDF in the random oracle version of the 1CCA attack game. Theorem 12.2. Assume H : X ! K is modeled as a random oracle. If T is oneway given an 0 is CCA secure. image oracle, and Es is 1CCA secure, then ETDF 453
0 In particular, for every 1CCA adversary A that attacks ETDF as in the random oracle version of Definition 12.2, there exist an inverting adversary Biow that breaks the onewayness assumption for T as in Attack Game 12.2, and a 1CCA adversary Bs that attacks Es as in Definition 9.6, where Biow and Bs are elementary wrappers around A, such that 0 1CCAro adv[A, ETDF ] 2 · IOWadv[Biow , T ] + 1CCAadv[Bs , Es ].
(12.3)
For applications of this theorem in the sequel, we record here some further technical properties that the adversary Biow satisfies. If A makes at most Qd decryption queries, then Biow makes at most Qd imageoracle queries. Also, the only dependence of Biow on the function F is that it invokes F (pk , ·) as a subroutine, at most Qro times, where Qro is a bound on the number of randomoracle queries made by A; ˆ, it always evaluates F (pk , ·) at x ˆ. moreover, if Biow produces an output x
Proof idea. The crux of the proof is to show that the adversary’s decryption queries do not help him in any significant way. What this means technically is that we have to modify the challenger so that it can compute responses to the decryption queries without using the secret key sk . The trick to achieve this is to exploit the fact that our challenger is in charge of implementing the random oracle, maintaining a table of all input/output pairs. Assume the target ciphertext (i.e., the one resulting from the encryption query) is (y, c), where y = F (pk , x), and suppose the challenger is given a decryption query (ˆ y , cˆ), where y 6= yˆ = F (pk , x ˆ). • If the adversary has previously queried the random oracle at x ˆ, and if kˆ was the output of ˆ the random oracle at x ˆ, then the challenger simply decrypts cˆ using k. • Otherwise, if the adversary has not made such a random oracle query, then the challenger does not know the correct value of the symmetric key — but neither does the adversary. The challenger is then free to choose a key kˆ at random, and decrypt cˆ using this key; however, the challenger must do some extra bookkeeping to ensure consistency, so that if the adversary ever queries the random oracle in the future at the point x ˆ, then the challenger “backpatches” ˆ the random oracle, so that its output at x ˆ is set to k. We also have to deal with decryption queries of the form (y, cˆ), where cˆ 6= c. Intuitively, under the onewayness assumption for T , the adversary will never query the random oracle at x, and so from the adversary’s point of view, the symmetric key k used in the encryption query, and used in 0 follows decryption queries of the form (y, cˆ), is as good as random, and so CCA security for ETDF immediately from 1CCA security for Es . In the above, we have ignored ciphertext queries of the form (ˆ y , cˆ) where yˆ has no preimage under F (pk , ·). The real decryption algorithm rejects such queries. This is why we need to assume T is oneway given an image oracle — in the reduction, we need this image oracle to reject ciphertexts of this form. 2 Proof. It is convenient to prove the theorem using the bitguessing versions of the 1CCA attack games. We prove: 0 ] IOWadv[Biow , T ] + 1CCAadv⇤ [Bs , Es ]. 1CCAro adv⇤ [A, ETDF
Then (12.3) follows by (12.2) and (9.2). 454
(12.4)
As usual, we define Game 0 to be the game played between A and the challenger in the bit0 . We then modify the challenger to guessing version of the 1CCA attack game with respect to ETDF obtain Game 1. In each game, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. Also, for j = 0, 1, we define Wj to be the event that ˆb = b in Game j. Game 0. The logic of the challenger is shown in Fig. 12.1. The challenger has to respond to random oracle queries, in addition to encryption and decryption queries. The adversary can make any number of random oracle queries, and any number of decryption queries, but at most one encryption query. Recall that in addition to direct access to the random oracle via explicit random oracle queries, the adversary also has indirect access to the random oracle via the encryption and decryption queries, where the challenger also makes use of the random oracle. In the initialization step, the challenger computes (pk , sk ) R G(); we also have our challenger make those computations associated with the encryption query that can be done without yet knowing the challenge plaintext. To facilitate the proof, we want our challenger to use the secret key sk as little as possible in processing decryption queries. This will motivate a somewhat nontrivial strategy for implementing the decryption and random oracle queries. As usual, we will make use of an associative array to implement the random oracle. In the proof of Theorem 11.2, which analyzed the semantic security of ETDF , we did this quite naturally by using an associative array Map : X ! K. We could do the same thing here, but because we want our challenger to use the secret key as little as possible, we adopt a di↵erent strategy. Namely, we will represent the random oracle using associative array Map 0 : Y ! K, with the convention that ˆ where yˆ = F (pk , x y ] = k, ˆ). We if the value of the oracle at x ˆ 2 X is equal to kˆ 2 K, then Map 0 [ˆ will also make use of an associative array Pre : Y ! X that is used to track explicit random oracle queries made by the adversary: if Pre[ˆ y] = x ˆ, this means that the adversary queried the oracle at the point x ˆ, and yˆ = F (pk , x ˆ). Note that Map 0 will in general be defined at points other than those at which Pre is defined, since the challenger also makes random oracle queries. In preparation for the encryption query, in the initialization step, the challenger precomputes x R X, y F (pk , x), k R K. It also sets Map 0 [y] k, which means that the value of the random oracle at x is equal to k. Also note that in the initialization step, the challenger sets c ?, and in processing the encryption query, overwrites c with a ciphertext in C. Thus, decryption queries processed while c = ? are phase 1 queries, while those processed while c 6= ? are phase 2 queries. To process a decryption query (ˆ y , cˆ), making minimal use of the secret key, the challenger uses the following strategy. • If yˆ = y, the challenger just uses the prepared key k directly to decrypt cˆ. • Otherwise, the challenger checks if Map 0 is defined at the point yˆ, and if not, it assigns to ˆ If yˆ has a preimage x y ] a random value k. ˆ and Map 0 was not defined at yˆ, this means Map 0 [ˆ that neither the adversary nor the challenger previously queried the random oracle at x ˆ, and so this new random value kˆ represents the value or the random oracle at x ˆ; in particular, if the adversary later queries the random oracle at the point x ˆ, this same value of kˆ will be 0 used. If yˆ has no preimage, then assigning Map [ˆ y ] a random value kˆ has no real e↵ect — it just streamlines the logic a bit. • Next, the challenger tests if yˆ is in the image of F (pk , ·). If yˆ is not in the image, the challenger just rejects the ciphertext. In Fig. 12.1, we implement this by invoking the function
455
initialization: F (pk , x) (pk , sk ) R G(), x R X , y c ? initialize empty associative arrays Pre : Y ! X and Map 0 : Y ! K k R K, b R {0, 1} k (1) Map 0 [y] send the public key pk to A; upon receiving an encryption query (m0 , m1 ) 2 M2 : b R {0, 1}, c R Es (k, mb ), send (y, c) to A;
upon receiving a decryption query (ˆ y , cˆ) 2 X ⇥ C, where (ˆ y , cˆ) 6= (y, c): if yˆ = y then m ˆ Ds (k, cˆ) else if yˆ 2 / Domain(Map 0 ) then Map 0 [ˆ y] R K (2) if Image(pk , sk , yˆ) = “no” // i.e., yˆ is not in the image of F (pk , ·) then m ˆ reject ˆ cˆ) y ], m ˆ Ds (k, else kˆ Map 0 [ˆ send m ˆ to A; upon receiving a random oracle query x ˆ 2 X: yˆ F (pk , x ˆ), Pre[ˆ y] x ˆ if yˆ 2 / Domain(Map 0 ) then Map 0 [ˆ y] R K 0 y ] to A send Map [ˆ
Figure 12.1: Game 0 challenger Image(pk , sk , yˆ). For now, we can think of Image as being implemented as follows: ⇢ Image(pk , sk , yˆ) := return “yes” if F (pk , I(sk , yˆ)) = yˆ and “no” otherwise
.
This is the only place where our challenger makes use of the secret key. • Finally, if yˆ is in the range of F (pk , ·), the challenger simply decrypts cˆ directly using the y ], which at this point is guaranteed to be defined, and represents symmetric key kˆ = Map 0 [ˆ the value of the random oracle at the preimage x ˆ of yˆ. Note that our challenger can do this, without actually knowing x ˆ. This is the crux of the proof. Despite this somewhat involved bookkeeping, it should be clear that our challenger behaves exactly as in the usual attack game. Game 1. This game is precisely the same as Game 0, except that we delete the line marked (1) in Fig. 12.1. Let Z be the event that the adversary queries the random oracle at x in Game 1. Clearly, 456
Games 0 and 1 proceed identically unless Z occurs, and so by the Di↵erence Lemma, we have Pr[W1 ]
Pr[W0 ] Pr[Z].
(12.5)
If event Z happens, then at the end of Game 1, we have Pre[y] = x. What we want to do, therefore, is use A to build an efficient adversary Biow that breaks the onewayness assumption for T with an advantage equal to Pr[Z], with the help of an image oracle. The logic of Biow is very straightforward. Basically, after obtaining the public key pk and y 2 Y from its challenger in Attack Game 12.2, Biow plays the role of challenger to A as in Game 1. The value of x is never explicitly used in that game (other than to compute y), and the value of the secret key sk is not used, except in the evaluation of the Image function, and for this, Biow can use the image oracle provided to it in Attack Game 12.2. At the end of the game, if y 2 Domain(Pre), then Biow outputs x = Pre[y]. It should be clear, by construction, that Pr[Z] = OWadv[Biow , T ].
(12.6)
Finally, note that in Game 1, the key k is only used to encrypt the challenge plaintext, and to process decryption queries of the form (y, cˆ), where cˆ 6= c. As such, the adversary is essentially just playing the 1CCA attack game against Es at this point. More precisely, we can easily derive an efficient 1CCA adversary Bs based on Game 1 that uses A as a subroutine, such that Pr[W1 ]
1/2 = 1CCAadv⇤ [Bs , Es ].
(12.7)
This adversary Bs generates (pk , sk ) itself and uses sk to answer queries from A. Combining (12.5), (12.6) and (12.7), we obtain (12.4). That completes the proof of the theorem. 2
12.3.1
0 Instantiating ETDF with RSA
0 Suppose we instantiate ETDF using RSA just as we did in Section 11.4.1. The underlying trapdoor function is actually a permutation on Zn . This implies two things. First, we can omit the check in the decryption algorithm that y is in the image of the trapdoor function, and so we end up with exactly the same scheme ERSA as was presented in Section 11.4.1. Second, the implementation of the image oracle in Attack Game 12.2 is trivial to implement, and so we end up back with Attack Game 10.2. Theorem 12.2 specializes as follows:
Theorem 12.3. Assume H : X ! K is modeled as a random oracle. If the RSA assumption holds for parameters (`, e), and Es is 1CCA secure, then ERSA is CCA secure. In particular, for every 1CCA adversary A that attacks ERSA as in the random oracle version of Definition 12.2, there exist an RSA adversary Brsa that breaks the RSA assumption for (`, e) as in Attack Game 10.3, and a 1CCA adversary Bs that attacks Es as in Definition 9.6, where Brsa and Bs are elementary wrappers around A, such that 1CCAro adv[A, ERSA ] 2 · RSAadv[Brsa , `, e] + 1CCAadv[Bs , Es ].
457
12.4
CCAsecure ElGamal encryption
We saw that the basic RSA encryption scheme ERSA could be shown to be CCA secure in the random oracle model under the RSA assumption (and assuming the underlying symmetric cipher was 1CCA secure). It is natural to ask whether the basic ElGamal encryption scheme EEG , discussed in Section 11.5, is CCA secure in the random oracle model, under the CDH assumption. Unfortunately, this is not the case: it turns out that a slightly stronger assumption than the CDH assumption is both necessary and sufficient to prove the security of EEG .
12.4.1
CCA security for basic ElGamal encryption
Recall that the basic ElGamal encryption scheme, EEG = (G, E, D), introduced in Section 11.5. It is defined in terms of a cyclic group G of prime order q generated by g 2 G, a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), and a hash function H : G ! K. The message space of EEG is M and the ciphertext space is G ⇥ C. Public keys are of the form u 2 G and secret keys are of the form ↵ 2 Zq . The algorithms G, E, and D are defined as follows: G()
:=
g ↵ , pk ↵ R Zq , u output (pk , sk );
E(u, m)
:=
Zq , v g , w output (v, c);
D(↵, (v, c) )
:=
w v↵ , k output m.
u, sk
R
u , k
H(w), m
↵ H(w), c
R
Es (k, m)
Ds (k, c)
To see why the CDH assumption by itself is not sufficient to establish the security of EEG against chosen ciphertext attack, suppose the public key is u = g ↵ . Now, suppose an adversary ˆ m) ˆ selects group elements vˆ and w ˆ in some arbitrary way, and computes kˆ H(w) ˆ and cˆ R Es (k, ⇤ for some arbitrary message m. ˆ Further, suppose the adversary can obtain the decryption m of the ciphertext (ˆ v , cˆ). Now, it is very likely that m ˆ = m⇤ if and only if w ˆ = vˆ↵ , or in other words, if and only if (u, vˆ, w) ˆ is a DHtriple. Thus, in the chosen ciphertext attack game, decryption queries can be e↵ectively used by the adversary to answer questions of the form “is (u, vˆ, w) ˆ a DHtriple?” for group elements vˆ and w ˆ of the adversary’s choosing. In general, the adversary would not be able to efficiently answer such questions on his own (this is the DDH assumption), and so these decryption queries may potentially leak some information about the secret key ↵. Based on the current state of our knowledge, this leakage does not seem to compromise the security of the scheme; however, we do need to state this as an explicit assumption. Intuitively, the interactive CDH assumption states that given a random instance (g ↵ , g ) of the DH problem, it is hard to compute g ↵ , even when given access to a “DHdecision oracle” that recognizes DHtriples of the form (g ↵ , ·, ·). More formally, this assumption is defined in terms of the following attack game. Attack Game 12.3 (Interactive Computational DiffieHellman). Let G be a cyclic group of prime order q generated by g 2 G. For a given adversary A, the attack game runs as follows. • The challenger computes ↵,
R
Zq , u
g↵, v
and gives (u, v) to the adversary. 458
g , w
g↵
• The adversary makes a sequence of DHdecision oracle queries to the challenger. Each query ˜ if so, is of the form (˜ v , w) ˜ 2 G2 . Upon receiving such a query, the challenger tests if v˜↵ = w; he sends “yes” to the adversary, and otherwise, sends “no” to the adversary. • Finally, the adversary outputs some w ˆ 2 G. We define A’s advantage in solving the interactive computational DiffieHellman problem, denoted ICDHadv[A, G], as the probability that w ˆ = w. 2 We stress that in the above attack game, the adversary can ask the challenger for help in determining whether certain triples are DHtriples, but only triples of the form (u, ·, ·), where u is generated by the challenger. Definition 12.4 (Interactive Computational DiffieHellman assumption). We say that the interactive computational DiffieHellman (ICDH) assumption holds for G if for all efficient adversaries A the quantity ICDHadv[A, G] is negligible. By the above discussion, we see (at least heuristically) that the ICDH assumption is necessary to establish the CCA security of EEG . Conversely, one can prove that EEG is CCA secure in the random oracle model under the ICDH assumption (and assuming also that Es is 1CCA secure); however, we shall instead analyze a slight variation of EEG , for which the reduction is simpler and 0 , is exactly the same as E more efficient. This encryption scheme, which we denote EEG EG , except that the symmetric key k is derived by hashing both v and w, instead of just w; that is, the hash function H is now of the form H : G2 ! K, and the symmetric key k is computed as k = H(v, w). 0 = (G, E, D) in its entirety. It is defined in terms For completeness, we describe the scheme EEG of a cyclic group G of prime order q generated by g 2 G, a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), and a hash function H : G2 ! K. Public keys are of the form u 2 G and secret keys are of the form ↵ 2 Zq . The algorithms G, E, and D are defined as follows: G()
:=
g ↵ , pk ↵ R Zq , u output (pk , sk );
E(u, m)
:=
Zq , v g , w output (v, c);
D(↵, (v, c) )
:=
w v↵ , k output m.
R
u, sk u , k
H(v, w), m
↵ H(v, w), c
R
Es (k, m)
Ds (k, c)
The message space is M and the ciphertext space is G ⇥ C. We have highlighted the di↵erences 0 between EEG and EEG . Theorem 12.4. Assume H : G2 ! K is modeled as a random oracle. If the ICDH assumption 0 is CCA secure. holds for G, and Es is 1CCA secure, then EEG 0 In particular, for every 1CCA adversary A that attacks EEG as in the random oracle version of Definition 12.2, there exist an ICDH adversary Bicdh for G as in Attack Game 12.3, and a 1CCA adversary Bs that attacks Es as in Definition 9.6, where Bicdh and Bs are elementary wrappers around A, such that 0 1CCAro adv[A, EEG ] 2 · ICDHadv[Bicdh , G] + 1CCAadv[Bs , Es ].
(12.8)
In addition, the number of DHdecision oracle queries made by Bicdh is bounded by the number of random oracle queries made by A.
459
Proof. The basic structure of the proof is very similar to that of Theorem 12.2. As in that proof, it is convenient to use the bitguessing versions of the 1CCA attack games. We prove 0 1CCAro adv⇤ [A, EEG ] ICDHadv[Bicdh , G] + 1CCAadv⇤ [Bs , Es ].
(12.9)
Then (12.8) follows by (12.2) and (9.2). We define Games 0 and 1. Game 0 is the bitguessing version of Attack Game 12.1 played by 0 . In each game, b denotes the random bit chosen by the challenger, while ˆ A with respect to EEG b ˆ denotes the bit output by A. For j = 0, 1, we define Wj to be the event that b = b in Game j. Game 0. The logic of the challenger is shown in Fig. 12.2. The adversary can make any number of random oracle queries, and any number of decryption queries, but at most one encryption query. As usual, in addition to direct access the random oracle via explicit random oracle queries, the adversary also has indirect access to the random oracle via the encryption and decryption queries, where the challenger also makes use of the random oracle. In the initialization step, the challenger computes the secret key ↵ 2 Zq and the public key u = g ↵ ; it also makes those computations associated with the encryption query that can be done without yet knowing the challenge plaintext. As in the proof of Theorem 12.2, we want our challenger to use the secret key ↵ as little as possible in processing decryption queries, and again, we use a somewhat nontrivial strategy for implementing the decryption and random oracle queries. Nevertheless, despite the significant superficial di↵erences, this implementation will be logically equivalent to the actual attack game. As usual, we will implement the random oracle using an associative array Map : G2 ! K. However, we will also make use of an auxiliary associative array Map 0 : G ! K. The convention ˆ then is that if (u, vˆ, w) ˆ is a DHtriple, and the value of the random oracle at the point (ˆ v , w) ˆ is k, 0 ˆ v ] = k. However, in processing a decryption query (ˆ v , cˆ), we may speculatively Map[ˆ v , w] ˆ = Map [ˆ assign a random value kˆ to Map 0 [ˆ v ], and then later, if the adversary queries the random oracle at the point (ˆ v , w), ˆ where (u, vˆ, w) ˆ is a DHtriple, we assign the value kˆ to Map[ˆ v , w], ˆ in order to maintain consistency. Now for more details. In preparation for the encryption query, in the initialization step, the R Zq , v g ,w g ↵ , k R K. It also sets Map[v, w] and Map 0 [v] to challenger precomputes k, which means that the value of the random oracle at (v, w) is equal to k. Also note that in the initialization step, the challenger sets c ?, and in processing the encryption query, overwrites c with a ciphertext in C. Thus, decryption queries processed while c = ? are phase 1 queries, while those processed while c 6= ? are phase 2 queries.
Processing random oracle queries. When processing a random oracle query (ˆ v , w), ˆ if Map[ˆ v , w] ˆ has not yet been defined, the challenger proceeds as follows. • First, it tests if (u, vˆ, w) ˆ is a DHtriple. In Fig. 12.2, we implement this by invoking the function DHP (↵, vˆ, w). ˆ For now, we can think of DHP as being implemented as follows: ˆ DHP (↵, vˆ, w) ˆ := vˆ↵ = w. This is the only place where our challenger makes use of the secret key. v ] to a random value, if it is not already • If (u, vˆ, w) ˆ is a DHtriple, the challenger sets Map 0 [ˆ 0 v ]. It also sets Sol [ˆ v] w, ˆ where Sol : G ! G is defined, and then sets Map[ˆ v , w] ˆ Map [ˆ another associative array. The idea is that Sol records solutions to DiffieHellman instances (u, vˆ) that are discovered while processing random oracle queries. 460
• If (u, vˆ, w) ˆ is not a DHtriple, then the challenger just sets Map[ˆ v , w] ˆ to a random value. The result of the random oracle query is always Map[ˆ v , w]. ˆ Processing decryption queries. In processing a decryption query (ˆ v , cˆ), the challenger proceeds as follows. • If vˆ = v, the challenger just uses the prepared key k directly to decrypt cˆ. • Otherwise, the challenger checks if Map 0 is defined at the point vˆ, and if not, it assigns to v ] a random value. It then uses the value kˆ = Map 0 [ˆ v ] directly to decrypt cˆ. Observe Map 0 [ˆ that our challenger performs the decryption without using the solution w ˆ to the instance (u, vˆ) of the CDH problem. However, if the adversary queries the random oracle at the point ˆ and so consistency is maintained. (ˆ v , w), ˆ the adversary will see the same value k, Hopefully, it is clear that our challenger behaves exactly as in the usual attack game, despite the more elaborate bookkeeping. Game 1. This game is the same as Game 0, except that we delete line (1) in Fig. 12.2. Let Z be the event that A queries the random oracle at (v, w) in Game 1. It is not hard to see that Games 0 and 1 proceed identically, unless Z occurs. By the Di↵erence Lemma, we have Pr[W1 ]
Pr[W0 ] Pr[Z].
(12.10)
If event Z happens, then at the end of Game 1, we have Sol [v] = w. What we want to do, therefore, is use A to build an efficient adversary Bicdh that breaks the CDH assumption for G, with the help of a DHdecision oracle, with an advantage equal to Pr[Z]. The logic of Bicdh is very straightforward. Basically, after obtaining u and v from its challenger in Attack Game 12.3, Bicdh plays the role of challenger to A as in Game 1. Besides the computation of u, the value of ↵ is never explicitly used in that game, other than in the evaluation of the DHP function, and for this, Bicdh can use the DHdecision oracle provided to it in Attack Game 12.3. At the end of the game, if v 2 Domain(Sol ), then Bicdh outputs w = Sol [v]. By construction, it is clear that Pr[Z] = ICDHadv[Bicdh , G].
(12.11)
Finally, note that in Game 1, the key k is only used to encrypt the challenge plaintext, and to process decryption queries of the form (v, cˆ), where cˆ 6= c. As such, the adversary is essentially just playing the 1CCA attack game against Es at this point. More precisely, we can easily derive an efficient 1CCA adversary Bs based on Game 1 that uses A as a subroutine, such that Pr[W1 ]
1/2 = 1CCAadv⇤ [Bs , Es ].
(12.12)
We leave the details of Bs to the reader. Combining (12.10), (12.11), and (12.12), we obtain (12.9). That completes the proof of the theorem. 2
461
initialization: g↵, v g ,w g↵ ↵, R Zq , u k R K, b R {0, 1} c ? initialize three empty associative arrays Map : G2 ! K, Map 0 : G ! K, and Sol : G ! G k (1) Map[v, w] k, Map 0 [v] send the public key u to A; upon receiving an encryption query (m0 , m1 ) 2 M2 : c R Es (k, mb ), send (v, c) to A;
upon receiving a decryption query (ˆ v , cˆ) 2 G ⇥ C, where (ˆ v , cˆ) 6= (v, c): if vˆ = v then m ˆ Ds (k, cˆ) else v] R K if vˆ 2 / Domain(Map 0 ) then Map 0 [ˆ ˆ cˆ) v ], m ˆ Ds (k, kˆ Map 0 [ˆ send m ˆ to A;
upon receiving a random oracle query (ˆ v , w) ˆ 2 G2 : if (ˆ v , w) ˆ 2 / Domain(Map) then if DHP(↵, vˆ, w) ˆ then v] if vˆ 2 / Domain(Map 0 ) then Map 0 [ˆ v ], Sol [ˆ v] w ˆ Map[ˆ v , w] ˆ Map 0 [ˆ else Map[ˆ v , w] ˆ R K send Map[ˆ v , w] ˆ to A
R
K
Figure 12.2: Game 0 challenger 0 is CCAsecure, in the random oracle model, under the ICDH Discussion. We proved that EEG assumption. Is the ICDH assumption reasonable? On the one hand, in Chapter 16 we will see groups G where the ICDH assumption is equivalent to the CDH assumption. In such groups there is no harm in assuming ICDH. On the other hand, the ElGamal system is most commonly implemented in groups where ICDH is not known to be equivalent to CDH. Is it reasonable to assume ICDH in such groups? Currently, we do not know of any group where CDH holds, but ICDH does not hold. As such, it appears to be a reasonable assumption to use when constructing cryptographic schemes. Later, in Section 12.6.2, we will see a variant of ElGamal encryption that is CCAsecure, in the random oracle model, under the normal CDH assumption.
462
12.5
CCA security from DDH without random oracles
In Section 11.5.2, we proved that EEG was semantically secure without relying on the random oracle model. Rather, we used the DDH assumption (among other assumptions). Unfortunately, it seems 0 , for that matter) is CCA secure without unlikely that we can ever hope to prove that EEG (or EEG relying on random oracles. In this section, we present a public key encryption scheme that can be proved CCA secure without relying on the random oracle heuristic. The scheme is based on the DDH assumption (as well as a few other standard assumptions). The scheme is a variant of one designed by Cramer and Shoup, and we call it ECS . It is built out of several components: • a cyclic group G of prime order q with generator g 2 G, • a symmetric cipher Es = (Es , Ds ), defined over (K, M, C), • a hash function H : G ! K, • a hash function H 0 : G ⇥ G ! Zq . The message space for ECS is M, and the ciphertext space is G3 ⇥ C. We now describe the key generation, encryption, and decryption algorithms for ECS . • the key generation algorithm runs as follows: G() :=
g↵ ↵ R Zq , u for i = 1, . . . , 3: i , ⌧i R Zq , ui g i u⌧i pk (u, u1 , u2 , u3 ), sk ( 1 , ⌧1 , 2 , ⌧2 , output (pk , sk );
3 , ⌧3 )
• for a given public key pk = (u, u1 , u2 , u3 ) 2 G4 and message m 2 M, the encryption algorithm runs as follows: E(pk , m) :=
Zq , v g , w u , ⇢ ⇢ w1 u1 , w 2 (u2 u3 ) k H(w1 ), c R Es (k, m) output (v, w, w2 , c); R
H 0 (v, w)
• for a given secret key sk = ( 1 , ⌧1 , 2 , ⌧2 , 3 , ⌧3 ) 2 Z6q and a ciphertext (v, w, w2 , c) 2 G3 ⇥ C, the decryption algorithm runs as follows: D(sk , (v, w, w2 , c) ) :=
⇢ H 0 (v, w) if v 2 +⇢ 3 w⌧2 +⇢⌧3 = w2 then w1 v 1 w ⌧1 , k else m reject output m.
H(w1 ), m
Ds (k, c)
We first argue that ECS satisfies the basic correctness property, i.e., that decryption undoes encryption. Consider an arbitrary encryption of a message m, which has the form (v, w, w2 , c), where v = g , w = u , ⇢ = H 0 (v, w), w1 = u1 , w2 = (u2 u⇢3 ) , k = H(w1 ), c = Es (k, m). 463
First, observe that v
2 +⇢ 3
w⌧2 +⇢⌧3 = g
(
2 +⇢ 3 )
u
(⌧2 +⇢⌧3 )
= (u2 u⇢3 ) = w2 .
This implies that the test in the decryption algorithm succeeds. Second, observe that v 1 w ⌧1 = g
1
u
⌧1
= u1 = w 1 .
This implies that the decryption algorithm derives the same symmetric key k as was used in encryption, and correctness for ECS follows from correctness for Es . We shall prove that ECS is CCA secure under the following assumptions: • the DDH assumption holds in G; • Es is 1CCA secure; • H is a secure KDF (see Definition 11.5); • H 0 is collision resistant (see Definition 8.1). One can in fact prove security of ECS under a weaker assumption on H 0 (namely, target collision resistance — see Definition 8.5). Moreover, a variation of ECS can be proved secure under an assumption that is somewhat weaker than the DDH assumption (namely, the Hash DiffieHellman assumption, discussed in Exercise 11.13). These results are developed below in the exercises. Theorem 12.5. If the DDH assumption holds in G, Es is 1CCA secure, H is a secure KDF, and H 0 is collision resistant, then ECS is CCA secure. In particular, for every 1CCA adversary A that attacks ECS as in Definition 12.2, and makes at most Qd decryption queries, there exist a DDH adversary Bddh for G as in Attack Game 10.6, a 1CCA adversary Bs that attacks Es as in Definition 9.6, a KDF adversary Bkdf that attacks H as in Attack Game 11.3, and a collisionfinding adversary Bcr that attacks H 0 as in Attack Game 8.1, where Bddh , Bs , Bkdf , Bcr are elementary wrappers around A, such that ⇣ 1CCAadv[A, ECS ] 2 DDHadv[Bddh , G] + KDFadv[Bkdf , H] (12.13) Qd + 1 ⌘ + 1CCAadv[Bs , Es ]. + CRadv[Bcr , H 0 ] + q
Proof. As usual, it is convenient to use the bitguessing versions of the 1CCA attack games. We prove 1CCAadv⇤ [A, ECS ] DDHadv[Bddh , G] + KDFadv[Bkdf , H] Qd + 1 + 1CCAadv⇤ [Bs , Es ]. + CRadv[Bcr , H 0 ] + q
(12.14)
Then (12.13) follows by (12.2) and (9.2). We define a series of games, Game j for j = 0, . . . , 6. Game 0 is the bitguessing version of Attack Game 12.1 played by A with respect to ECS . In each game, b denotes the random bit chosen by the challenger, while ˆb denotes the bit output by A. For j = 0, . . . , 6, we define Wj to be the event that ˆb = b in Game j. 464
(1)
(2) (3) (4)
initialization: ↵, R Zq ↵ g ,w g u g↵, v 0 ⇢ H (v, w) g i u⌧i for i = 1, . . . , 3: i , ⌧i R Zq , ui w1 u1 w2 (u2 u⇢3 ) k H(w1 ) ? b R {0, 1}, c send the public key (u, u1 , u2 , u3 ) to A;
upon receiving an encryption query (m0 , m1 ) 2 M2 : c R Es (k, mb ), send (v, w, w2 , c) to A;
upon receiving a decryption query (ˆ v , w, ˆ w ˆ2 , cˆ) 2 G3 ⇥ C, where (ˆ v , w, ˆ w ˆ2 , cˆ) 6= (v, w, w2 , c): if (ˆ v , w, ˆ w ˆ2 ) = (v, w, w2 ) then m ˆ Ds (k, cˆ) else ⇢ˆ H 0 (ˆ v , w) ˆ +ˆ ⇢ 2 3 w ˆ ⌧2 +ˆ⇢⌧3 = w ˆ2 then (5) if vˆ vˆ 1 w ˆ ⌧1 (6) w ˆ1 ˆ cˆ) ˆ Ds (k, kˆ H(w ˆ1 ), m else m ˆ reject send m ˆ to A.
Figure 12.3: Game 0 challenger Game 0. The logic of the challenger is shown in Fig. 12.3. The adversary can make any number of decryption queries, but at most one encryption query. Note that in the initialization step, the challenger performs those computations associated with the encryption query that it can, without yet knowing the challenge plaintext. Also note that in the initialization step, the challenger sets c ?, and in processing the encryption query, overwrites c with a ciphertext in C. Thus, decryption queries processed while c = ? are phase 1 queries, while those processed while c 6= ? are phase 2 queries. Game 1. We replace the lines marked (2) and (3) in Fig. 12.3 as follows: (2) (3)
w1 w2
v 1 w ⌧1 v 2 +⇢ 3 w⌧2 +⇢⌧3
Basically, we have simply replaced the formulas used to generate w1 and w2 in the encryption procedure with those used in the decryption procedure. As we already argued above in analyzing 465
the correctness property for ECS , these formulas are equivalent. In particular: Pr[W1 ] = Pr[W0 ].
(12.15)
The motivation for making this change is that now, the only place where we use the exponents ↵, , and is in the definition of the group elements u, v, and w, which allows us to then play the “DDH card” in the next step of the proof. Game 2. We replace the line marked (1) in Fig. 12.3 with R
(1)
Zq
After this change, the lines marked (1), (2), and (3) in Fig. 12.3 now read as follows: (1) (2) (3)
R
w1 w2
Zq v 1 w ⌧1 v 2 +⇢ 3 w⌧2 +⇢⌧3
It is easy to see that Pr[W1 ]
Pr[W2 ] DDHadv[Bddh , G]
(12.16)
for an efficient DDH adversary Bddh , which works as follows. After it obtains its DDH problem instance (u, v, w) from its own challenger, adversary Bddh plays the role of challenger to A in Game 0, but using the given values u, v, w. If (u, v, w) is a random DHtriple, then this is equivalent to Game 0, and if (u, v, w) is a random triple, this is equivalent to Game 1. At the end of the game, Bddh outputs 1 if ˆb = b and 0 otherwise. Game 3. We replace the line marked (1) in Fig. 12.3 with R
(1)
Zq \ {↵ }
After this change, the lines marked (1), (2), and (3) in Fig. 12.3 now read as follows: (1) (2) (3)
R
w1 w2
Zq \ {↵ } v 1 w ⌧1 v 2 +⇢ 3 w⌧2 +⇢⌧3
Since the statistical distance between the uniform distribution on all triples and the uniform distribution on all nonDHtriples is 1/q (see Exercise 10.6), it follows that: Pr[W2 ]
1 Pr[W3 ] . q
(12.17)
Interlude. Before continuing with the proof, let us see what the changes so far have accomplished. Consider any fixed values of ↵, , and 6= ↵ . Moreover, consider the group elements u1 , w1 generated by the challenger. These satisfy the equations u 1 = g 1 u⌧1 = g
1 +↵⌧1
and
w 1 = v 1 w ⌧1 = g
Taking discrete logarithms, we can write this as a matrix equation ◆ ✓ ◆✓ ◆ ✓ Dlogg u1 1 ↵ 1 = . Dlogg w1 ⌧1  {z } =:M
466
1+
⌧1
.
(12.18)
Now, the matrix M is nonsingular. One way to see this is to calculate its determinant det(M ) = ↵ 6= 0. Another way to see this is to observe that the second row of M cannot be a scalar multiple of the first: if it were, then by looking at the first column of M , the second row of M would have to be equal to times the first, and by looking at the second column of M , this would imply = ↵ , which is not the case. Next, observe that 1 and ⌧1 are uniformly and independently distributed over Zq . Since M is nonsingular, it follows from (12.18) that Dlogg u1 and Dlogg w1 are also uniformly and independently distributed over Zq . Equivalently, u1 and w1 are uniformly and independently distributed over G. If the adversary does not submit any decryption oracle queries, he learns nothing more about u1 and w1 , and since w1 is only used to derive the key k and then encrypt mb , security follows easily from the assumptions that H is a secure KDF and Es is semantically secure. Unfortunately, if the adversary does make decryption queries, these could potentially leak inv , w, ˆ w ˆ2 , cˆ) such that formation about w1 . Specifically, suppose the adversary submits a ciphertext (ˆ ˆ ⌧1 computed (u, vˆ, w) ˆ is not a DHtriple, yet passes the test at line (5). Then the value of w ˆ1 = vˆ 1 w on line (6), together with the value u1 in the public key, completely determine the values of 1 and ⌧1 , and hence the value of w1 . This can be seen by again considering a matrix equation as above. ˆ with ˆ 6= ↵ ˆ, then Indeed, if ˆ := Dlogg vˆ and ˆ = Dlogg w, ✓
Dlogg u1 Dlogg w ˆ1
◆
✓
◆ ✓ ◆ 1 ↵ = ˆ . 1 . ⌧1 ˆ  {z } c =:M
c is nonsingular, and so the values Dlogg u1 and Dlogg w Again, the matrix M ˆ1 completely determine 1 and ⌧1 . So to complete the proof, we shall argue that with overwhelming probability, the scenario described in the previous paragraph does not occur. That is, we shall argue that whenever the adversary submits a ciphertext (ˆ v , w, ˆ w ˆ2 , cˆ), where (u, vˆ, w) ˆ is not a DHtriple, the test at line (5) will pass with only negligible probability. That is the point of including the extra group elements u2 and u3 in the public key and the extra group element w2 in the ciphertext. Game 4. This is the same as Game 3, except we replace lines (5) and (6) by (5) (6)
ˆ and vˆ↵2 +ˆ⇢↵3 = w ˆ2 then if vˆ↵ = w ↵ 1 vˆ w ˆ1
where we define ↵i :=
i
+ ↵⌧i
(i = 1, . . . , 3).
(12.19)
Observe that if (u, vˆ, w) ˆ is not a DHtriple, then the modified test in line (5) will not pass; otherwise, ˆ one can verify that this test passes if and only if the original test in if it is a DHtriple (i.e., vˆ↵ = w), Game 3 passes, and the computation of w ˆ1 on line (6) is equivalent to that in Game 3. In particular, this new test is strictly stronger than the test in Game 3. Also notice that the computations in lines (5) and (6) in Game 4 do not depend directly on the individual values of 1 , ⌧1 , 2 , ⌧2 , 3 , and ⌧3 , but rather, only indirectly, via the values ↵1 , ↵2 , and ↵3 , defined in (12.19). The importance of this will become evident later in the proof. After this change, the lines marked (1), (2), (3), (5), and (6) in Fig. 12.3 now read as follows:
467
(1) (2) (3) (5) (6)
R
w1 w2
Zq \ {↵ } v 1 w ⌧1 v 2 +⇢ 3 w⌧2 +⇢⌧3 if vˆ↵ = w ˆ and vˆ↵2 +ˆ⇢↵3 = w ˆ2 then ↵ 1 w ˆ1 vˆ
Define Z to be the event that in Game 4, for some decryption query, the test in line (5) is ˆ2 = w2⇤ , where performed, and we have w ˆ 6= vˆ↵ but w w2⇤ := vˆ
⇢ 3 2 +ˆ
w ˆ ⌧2 +ˆ⇢⌧3 .
Such a ciphertext is rejected in Game 4, but not in Game 3. However, the two games proceed identically unless Z occurs, and so by the Di↵erence Lemma, we have Pr[W3 ]
Pr[W4 ] Pr[Z].
(12.20)
To bound Pr[Z], it will also be convenient to consider the event Z 0 that for the relevant decryption query, we have (v, w) 6= (ˆ v , w) ˆ but H 0 (v, w) = H 0 (ˆ v , w), ˆ that is, (v, w) and (ˆ v , w) ˆ form a 0 collision under H . Clearly, we have Pr[Z] Pr[Z 0 ] + Pr[¬Z 0 ^ Z].
(12.21)
Pr[Z 0 ] CRadv[Bcr , H 0 ]
(12.22)
It should be clear that for an efficient collisionfinding adversary Bcr . Indeed, adversary Bcr just plays Game 4 and waits for the event Z 0 to happen. So now we are left to bound Pr[¬Z 0 ^ Z]. We claim that Pr[¬Z 0 ^ Z]
Qd , q
(12.23)
where Qd is an upper bound on the number of decryption queries. To prove (12.23), it will suffice to consider the event ¬Z 0 ^ Z for just a single decryption query and apply the union bound. So consider a fixed decryption query (ˆ v , w, ˆ w ˆ2 , cˆ), and suppose that ¬Z 0 ^Z occurs at this query. We must have (ˆ v , w, ˆ w ˆ2 ) 6= (v, w, w2 ), as otherwise, we would not even reach the test at line (5). ˆ2 , and so event Z could not have We must also have (ˆ v , w) ˆ 6= (v, w), as otherwise w2⇤ = w2 6= w occurred at this query. Moreover, since Z 0 does not occur at this query, we must have ⇢ˆ 6= ⇢. Let ˆ := Dlog vˆ and ˆ = Dlog w. ˆ g g ˆ Since Z occurs at this query, we must have ˆ 6= ↵ . 0 Summarizing, if ¬Z ^ Z occurs at this query, we must have ⇢ˆ 6= ⇢,
ˆ 6= ↵ ˆ,
and
w ˆ2 = w2⇤ .
We can express the relationship between the values 2 , ⌧2 , 3 , ⌧3 and the values Dlogg u2 , Dlogg u3 , Dlogg w2 , Dlogg w2⇤ as a matrix equation: 0 1 0 10 1 1 ↵ 0 0 Dlogg u2 2 B Dlogg u3 C B 0 0 1 ↵ C B ⌧2 C B C B CB C. (12.24) @Dlogg w2 A = @ ⇢ ⇢ A @ 3A ˆ ˆ ⇢ˆ ˆ ⇢ˆˆ ⌧3 Dlogg w2⇤ {z }  =:M
468
An essential fact is that the matrix M is nonsingular. Indeed, one can again just compute the determinant ↵ )(ˆ ↵ ˆ), det(M ) = (⇢ ⇢ˆ)( which is nonzero under our assumptions. Since 2 , ⌧2 , 3 , and ⌧3 are uniformly and independently distributed over Zq , and M is nonsingular, the values Dlogg u2 , Dlogg u3 , Dlogg w2 , and Dlogg w2⇤ are also uniformly and independently distributed over Zq . Moreover, in Game 4, the only information the adversary obtains about 2 , ⌧2 , 3 , and ⌧3 is that implied by the values Dlogg u2 , Dlogg u3 , and Dlogg w2 . This is where we use the fact that the test at line (5) is now implemented in terms of the values ↵2 = Dlogg u2 and ↵3 = Dlogg u3 , defined in (12.19). That is, the test itself only uses information that is already present in the public key. It follows that the value w ˆ2 computed by the adversary is independent ˆ2 = w2⇤ with probability 1/q. The bound (12.23) then follows of the correct value w2⇤ ; therefore, w from the union bound. Game 5. We replace the line marked (2) with (2)
R
w1
G
After this change, the lines marked (1), (2), (3), (5), and (6) in Fig. 12.3 now read as follows: (1) (2) (3) (5) (6)
R
Zq \ {↵ } G v 2 +⇢ 3 w⌧2 +⇢⌧3 if vˆ↵ = w ˆ and vˆ↵2 +ˆ⇢↵3 = w ˆ2 then vˆ↵1 w ˆ1
R
w1 w2
We claim that Pr[W5 ] = Pr[W4 ].
(12.25)
This is because, as already argued in the analysis of Game 2, the values Dlogg u1 and Dlogg w1 are related to the random values 1 and ⌧1 by the matrix equation (12.18), where the matrix M is nonsingular. Moreover, in Game 4, the only information the adversary obtains about 1 and ⌧1 is that implied by Dlogg u1 and Dlogg w1 . This is where we use the fact that the computation at line ˆ1 at line (6) only uses (6) is implemented in terms of ↵1 = Dlogg u1 . That is, the computation of w information that is already present in the public key. Thus, replacing w1 by a truly random group element does not really change the game at all. Game 6. Finally, the stage is set to play our “KDF card” and “1CCA card”. We replace the line marked (4) by (4)
k
R
K
After this change, the lines marked (1)–(6) in Fig. 12.3 now read as follows: (1) (2) (3) (4) (5) (6)
Zq \ {↵ } w1 R G v 2 +⇢ 3 w⌧2 +⇢⌧3 w2 R k K ˆ and vˆ↵2 +ˆ⇢↵3 = w ˆ2 then if vˆ↵ = w ↵ 1 w ˆ1 vˆ R
469
It should be clear that Pr[W5 ]
Pr[W6 ] KDFadv[Bkdf , H]
(12.26)
1/2 = 1CCAadv⇤ [Bs , Es ],
(12.27)
and Pr[W6 ]
where Bkdf is an efficient adversary attacking H as a KDF, and Bs is a 1CCA adversary attacking Es . The bound (12.14) now follows directly from (12.15), (12.16), (12.17), (12.20), (12.21), (12.22), (12.23), (12.25), (12.26), and (12.27). That completes the proof of the theorem. 2
12.6
CCA security via a generic transformation
We have presented several constructions of CCAsecure public key encryption schemes. In Section 12.3, we saw how to achieve CCA security in the random oracle model using a trapdoor function scheme, and in particular (in Section 12.3.1) with RSA. In Section 12.4, we saw how to achieve CCA security in the random oracle model under the interactive CDH assumption, and with a bit more e↵ort, we were able to achieve CCA security in Section 12.5 without resorting to the random oracle model, but under the DDH assumption. It is natural to ask if there is a generic transformation that converts any CPAsecure public key encryption scheme into one that is CCAsecure, as we did for symmetric encryption in Chapter 9. The answer is yes. In the random oracle model it is possible to give a simple and efficient transformation from CPAsecurity to CCAsecurity. This transformation, called the FujisakiOkamoto transformation, allows one to efficiently convert any publickey encryption scheme that satisfies a very weak security property (weaker than CPA security) into a publickey encryption scheme that is CCAsecure in the random oracle model. It is possible, in principle, to give a similar transformation without relying on random oracles, however, the known constructions are too inefficient to be used in practice [33]. Applications. We show in Section 12.6.2 that applying the FujisakiOkamoto transformation to a variant of ElGamal encryption, gives a public key encryption scheme that is CCAsecure in the random oracle model under the ordinary CDH assumption, rather than the stronger, interactive CDH assumption. (Exercise 12.23 develops another approach to achieving the same result, with a tighter security reduction to the CDH assumption). Beyond ElGamal, the FujisakiOkamoto transformation can be applied to other public key encryption schemes, such as Regev’s latticebased encryption scheme discussed in Chapter 17, the McEliece codingbased scheme [73], and the NTRU scheme [54]. All these systems can be made CCA secure, in the random oracle model, using the technique in this section. The FujisakiOkamoto transformation. It is best to understand the FujisakiOkamoto transformation as a technique that allows us to build a trapdoor function scheme TFO that is one way, even given an image oracle (as in Definition 12.3), starting from any oneway, probabilistic public0 key encryption scheme Ea = (Ga , Ea , Da ). We can then plug TFO into the construction ETDF presented in Section 12.3, along with a 1CCA symmetric cipher, to obtain a publickey encryption scheme EFO that is CCA secure in the random oracle model. 470
Let Ea = (Ga , Ea , Da ) be an arbitrary publickey encryption scheme with message space X and ciphertext space Y. • The encryption algorithm Ea may be probabilistic, and in this case, it will be convenient to make its random coin tosses explicit. To this end, let us view Ea as a deterministic algorithm that takes three inputs: a public key pk , a message x 2 X , and a randomizer r 2 R, where R is some finite randomizer space. To encrypt a message x 2 X under a public key pk , one chooses r 2 R at random, and then computes the ciphertext Ea (pk , x; r). • In general, the decryption algorithm Da may return the special symbol reject; however, we will assume that this is not the case. That is, we will assume that Da always returns an element in the message space X . This is not a serious restriction, as we can always modify the decryption algorithm so as to return some default message instead of reject. This assumption will simplify the presentation somewhat. The FujisakiOkamoto transformation applied to Ea = (Ga , Ea , Da ) works as follows. We will also need a hash function U : X ! R, mapping messages to randomizers, which will be modeled as a random oracle in the security analysis. The trapdoor function scheme is TFO = (Ga , F, Da ), defined over (X , Y), where (12.28) F (pk , x) := Ea (pk , x; U (x)). To prove that TFO is one way given an image oracle, in addition to modeling U as a random oracle, we will need to make the following assumptions, which will be made more precise below: 1. Ea is one way, which basically means that given an encryption of a random message x 2 X , it is hard to compute x; 2. Ea is unpredictable, which basically means that a random reencryption of any ciphertext y 2 Y is unlikely to be equal to y. We now make the above assumptions more precise. As usual, the onewayness property is defined in terms of an attack game. Attack Game 12.4 (Oneway encryption). For a given publickey encryption scheme Ea = (Ga , Ea , Da ) with message space X , ciphertext space Y, and randomizer space R, and a given adversary A, the attack game proceeds as follows: • The challenger computes (pk , sk )
R
Ga (), x
R
X, r
R
R, y
Ea (pk , r; s),
and sends (pk , y) to the adversary. • The adversary outputs x ˆ 2 R. We say A wins the above game if x ˆ = x, and we define A’s advantage OWadv[A, Ea ] to be the probability that A wins the game. 2 Definition 12.5 (Oneway encryption). A publickey encryption scheme Ea is one way if for every efficient adversary A, the value OWadv[A, Ea ] is negligible. 471
Note that because Ea may be probabilistic, an adversary that wins Attack Game 12.4 may not even know that they have won the game. We define unpredictable encryption as follows. Definition 12.6 (Unpredictable encryption). Let Ea = (Ga , Ea , Da ) be a given publickey encryption scheme with message space X , ciphertext space Y, and randomizer space R. We say Ea is ✏unpredictable if for every possible output (pk , sk ) of Ga and every y 2 Y, if we choose r 2 R at random, then we have Pr[Ea (pk , Da (sk , y); r) = y] ✏. We say Ea is unpredictable if it is ✏unpredictable for negligible ✏. We note that the onewayness assumption is implied by semantic security (see Exercise 12.9). We also note that, any publickey encryption scheme that is semantically secure typically is also unpredictable, even though this is not implied by the definition. Moreover, any publickey encryption scheme can be easily transformed into one that satisfies this assumption, without a↵ecting the onewayness assumption (see Exercise 12.10). Theorem 12.6. If U is modeled as a random oracle, and if Ea is one way and unpredictable, then the trapdoor function scheme TFO , resulting from the FujisakiOkamoto transformation (12.28), is one way given an image oracle. In particular, assume that Ea is ✏unpredictable. Also assume that adversary A attacks TFO as in the random oracle version of Attack Game 12.2, and makes at most Qio image oracle queries and Qro random oracle queries. Moreover, assume that A always includes its output among its random oracle queries. Then there exists an adversary Bow that breaks the onewayness assumption for Ea as in Attack Game 12.4, where Bow is an elementary wrapper around A, such that (12.29) OWro adv[A, TFO ] Qio · ✏ + Qro · OWadv[B, Ea ].
Proof. We define Game 0 to be the game played between A and the challenger in the random oracle version of Attack Game 12.2 with respect to TFO = (Ga , F, Da ). We then modify the challenger several times to obtain Games 1, 2, and so on. In each game, x denotes the random element of X chosen by the challenger. For j = 0, 1, . . . , we define Wj to be the event that x is among the random oracle queries made by A in Game j. As stated above, we assume that A always queries the random oracle at its output value: this is a reasonable assumption, and we can always trivially modify an any adversary to ensure that it behaves this way, increasing its randomoracle queries by at most 1. Clearly, we have OWro adv[A, TFO ] Pr[W0 ]. (12.30) Game 0. The challenger in Game 0 has to respond to random oracle queries, in addition to image oracle queries. We make use of an associative array Map : X ! R to implement the random oracle representing the hash function U . The logic of the challenger is shown in Fig. 12.4. The adversary can make any number of random oracle queries and any number of image queries. The associative array Pre : Y ! X is used to track the adversary’s random oracle queries. Basically, Pre[ˆ y] = x ˆ means that yˆ is the image of x ˆ under F (pk , ·). Game 1. In this game, we make the following modification to the challenger. The line marked (2) in the logic for processing decryption queries is modified as follows:
472
initialization: Ea (pk , x; r) (pk , sk ) R Ga (), x R X , r R R, y initialize empty associative arrays Map : X ! R and Pre : Y ! X (1) Map[x] r send the public key pk to A;
upon receiving an image oracle query yˆ 2 Y: if yˆ = y then result “yes” else x ˆ Da (sk , yˆ) if x ˆ2 / Domain(Map) then Map[ˆ x] rˆ Map[ˆ x] ˆ; rˆ) = yˆ (2) if Ea (pk , x then result “yes” else result “no” send result to A;
R
X
upon receiving a random oracle query x ˆ 2 X: if x ˆ2 / Domain(Map) then Map[ˆ x] R R ˆ; rˆ), Pre[ˆ y] x ˆ rˆ Map[ˆ x], yˆ Ea (pk , x send rˆ to A
Figure 12.4: Game 0 challenger (2)
if yˆ 2 Domain(Pre)
Let Z1 be the event that in Game 1, the adversary submits an image oracle query yˆ such that yˆ 6= y,
yˆ 2 / Domain(Pre),
and
ˆ; rˆ) = yˆ, Ea (pk , x
where x ˆ and rˆ are computed as in the challenger. It is clear that Games 0 and 1 proceed identically unless Z1 occurs, and so by the Di↵erence Lemma, we have Pr[W1 ]
Pr[W0 ] Pr[Z1 ].
(12.31)
We argue that Pr[Z1 ] Qio · ✏,
(12.32)
where we are assuming that Ea is ✏unpredictable. Indeed, observe that in Game 1, if A makes an image query yˆ with yˆ 6= y and yˆ 2 / Domain(Pre), then either ˆ; rˆ) = y 6= yˆ with certainty, or • x ˆ = x, and so Ea (pk , x 473
upon receiving an image oracle query yˆ 2 Y: if yˆ 2 {y} [ Domain(Pre) then then result “yes” else result “no” send result to A
Figure 12.5: Modified logic for image oracle queries ˆ; rˆ) = yˆ with • x ˆ 6= x, and so rˆ is independent of A’s view, from which it follows that Ea (pk , x probability at most ✏. The inequality (12.32) then follows by the union bound. Game 2. This game is the same Game 1, except that we implement the image oracle queries using the logic described in Fig. 12.5. The idea is that in Game 1, we do not really need to use the secret key to implement the image oracle queries. It should be clear that (12.33) Pr[W2 ] = Pr[W1 ]. Since we do not use the secret key at all in Game 2, this makes it easy to play our “onewayness card.” Game 3. In this game, we delete the line marked (1) in Fig. 12.4. We claim that Pr[W3 ] = Pr[W2 ].
(12.34)
Indeed, Games 2 and 3 proceed identically until A queries the random oracle at x. So if W2 does not occur, neither does W3 , and if W3 does not occur, neither does W2 . That is, W2 and W3 are identical events. We sketch the design an efficient adversary B such that Pr[W3 ] Qro · OWadv[B, Ea ].
(12.35)
The basic idea, as usual, is that B plays the role of challenger to A, as in Game 3, except that the values pk , sk , x, r, and y are generated by B’s OW challenger, from which B obtains the values pk and y. Adversary B interacts with A just as the challenger in Game 3. The key observation is that B does not need to know the values sk , x, and r in order to carry out its duties. At the end of the game, if A made a random oracle query at the point x, then the value x will be contained in the set Domain(Map). In general, it may not be easy to determine which of the values in this set is the correct decryption of y, and so we use our usual guessing strategy; namely, B simply chooses an element at random from Domain(Map) as its guess at the decryption of y. It is clear that the inequality (12.35) holds. The inequality (12.29) now follows from (12.30)–(12.35). That proves the theorem. 2
474
12.6.1
A generic instantiation
Putting all the pieces together, we get the following publickey encryption scheme EFO . The components consist of: • a publickey encryption scheme Ea = (Ga , Ea , Da ), with message space X , ciphertext space Y, and randomizer space R; • a symmetric cipher Es = (Es , Ds ), with key space K and message space M; • hash functions H : X ! K and U : X ! R. The scheme EFO = (Ga , E, D) has message space M and ciphertext space Y ⇥ C. Encryption and decryption work as follows: E(pk , m)
:=
U (X), y Ea (pk , x; r) x R X, r k H(x), c R Es (k, m) output (y, c);
D(sk , (y, c) )
:=
x I(sk , y), r U (x) if Ea (pk , x; r) 6= y then m reject else k H(x), m output m.
Ds (k, c)
Combining Theorem 12.2 and Theorem 12.6, we immediately get the following: Theorem 12.7. If H and U are modeled as a random oracles, Ea is one way and unpredictable, and Es is 1CCA secure, then the above publickey encryption scheme EFO is CCA secure. In particular, assume that that Ea is ✏unpredictable. Then for every 1CCA adversary A that attacks EFO as in the random oracle version of Definition 12.2, and which makes at most Qd decryption queries, QH queries to the random oracle for H, and QU queries to the random oracle for U , there exist an adversary Bow that breaks the onewayness assumption for Ea as in Attack Game 12.4, and a 1CCA adversary Bs that attacks Es as in Definition 9.6, where Bow and Bs are elementary wrappers around A, such that 1CCAro adv[A, EFO ] 2(QH + QU ) · OWadv[Bow , Ea ] + 2Qd · ✏ + 1CCAadv[Bs , Es ].
12.6.2
(12.36)
A concrete instantiation with ElGamal
In the FujisakiOkamoto transformation, we can easily use a variant of ElGamal encryption in the role of Ea . Let G be a cyclic group of prime order q generated by g 2 G. We define a publickey encryption scheme Ea = (Ga , Ea , Da ), with message space G, ciphertext space G2 , and randomizer space Zq . Public keys are of the form u 2 G and secret keys of the form ↵ 2 Zq . Key generation, encryption, and decryption work as follows: Ga ()
:=
↵ R Zq , u g ↵ , pk output (pk , sk );
Ea (u, x; )
:=
v g , w u , y output (v, y);
Da (↵, (v, y))
:=
w v↵ , x output x.
y/w
475
u, sk wx
↵
We called this scheme multiplicative ElGamal in Exercise 11.5, where we showed that it is semantically secure under the DDH assumption. It easily verified that Ea has the following properties: • Ea is oneway under the CDH assumption. Indeed, an adversary A that breaks the onewayness assumption for Ea is easily converted to an adversary B that breaks the CDH with same advantage. Given an instance (u, v) 2 G2 of the CDH problem, adversary B plays the role of challenger against A in Attack Game 12.4 as follows: – B sets y
R
G, and gives A the public key u and the ciphertext (v, y);
– when A outputs x 2 G, adversary B outputs w
y/x.
Clearly, if x is the decryption of (v, y), then w = y/x is the solution to the given instance (u, v) of the CDH problem. • Ea is 1/qunpredictable. Moreover, under the CDH assumption, it must be the case that 1/q is negligible. EG = (G, E, D). Putting all the pieces together, we get the following publickey encryption scheme EFO The components consist of:
• a cyclic group G of prime order q generated by g 2 G; • a symmetric cipher Es = (Es , Ds ), with key space K and message space M; • hash functions H : G ! K and U : G ! Zq . EG is M and its ciphertext space is G2 ⇥C. Public keys are of the form u 2 G The message space of EFO and secret keys of the form ↵ 2 Zq . The key generation, encryption, and decryption algorithms work as follows:
G() :=
:=
g ↵ , pk ↵ R Zq , u output (pk , sk );
E(u, m)
:=
x R G, U (x), v g , w R Es (k, m) k H(x), c output (v, y, c);
D(↵, (v, y, c))
:=
w v↵ , x if g = v then k else m output m.
y/w, H(x), m reject
u, sk
↵ u , y
w·x
U (x) Ds (k, c)
Here, we have optimized the decryption algorithm a bit: if v = g , then it follows that Ea (pk , x; ) = (g , u x) = (v, y), and so it is unnecessary to execute all of algorithm Ea . As a special case of Theorem 12.7, we get the following: Theorem 12.8. If H and U are modeled as a random oracles, the CDH assumption holds for G, EG is CCA secure. and Es is 1CCA secure, then the above publickey encryption scheme EFO
476
EG In particular, for every 1CCA adversary A that attacks EFO as in the random oracle version of Definition 12.2, and which makes at most Qd decryption queries, QH queries to the random oracle for H, and QU queries to the random oracle for U , there exist an adversary Bcdh that breaks the CDH assumption for G as in Attack Game 10.5, and a 1CCA adversary Bs that attacks Es as in Definition 9.6, where Bcdh and Bs are elementary wrappers around A, such that EG ] 2(QH + QU ) · CDHadv[Bcdh , G] + 2Qd /q + 1CCAadv[Bs , Es ]. 1CCAro adv[A, EFO
(12.37)
Contrast this result to the construction in Section 12.4.1: to achieve CCA security, instead of the ordinary CDH assumption, that scheme requires the stronger, interactive CDH assumption.
12.7
CCAsecure publickey encryption with associated data
In Section 9.6, we introduced the notion of CCA security for symmetrickey ciphers with associated data. In this section, we briefly sketch how this notion can be adapted to publickey encryption. First, we have to deal with the syntactic changes. A publickey encryption scheme E = (G, E, D) with associated data, or AD publickey encryption scheme, has the same basic structure as an ordinary publickey encryption scheme, except that the encryption algorithm E and decryption algorithm D each take an additional input d, called the associated data. Thus, E gets invoked as c R E(pk , m, d), and D gets invoked as m D(sk , c, d). As usual, we require that ciphertexts generated by E are correctly decrypted by D, as long as both are given the same associated data. That is, for all possible outputs (pk , sk ) of G, and all messages m and associated dat